This is a question I think gets asked a lot as there’s no doubt some confusion with how some gear skills affect certain character skills (if at all) given the variety of characters and classes. I had a look around at the data to try and come up with a better understanding of it all so that I could put together another chart (at the request of @jjand302) for quick reference. Unfortunately this is a very complicated topic that players will also no doubt contest how certain effects are applied. So treat this as an introduction into a conversation where we can debunk all the myths and put together an accurate set of information that everyone can use. 
Disclaimer: This information was put together using a combination of information I have data mined and my own personal assumptions based on what I have read online or played in the game. To date I have also only played 16 23 out of the 26 27 playable characters so I have very little experience with the other 10 4 characters being; Alani, Boldur, Attikus, Deande, El Dragon, Kleese, Galilea, Ghalt, Reyna, Shayne & Aurox and Pendles. I have no doubt there will be errors, which is why this is a work in progress. If you have a really good understanding of a character, I want your input in this discussion!!
Here’s a work in progress chart that shows which character skills/weapons are affected/related by what gear skills. For version 1, I have mainly focused on weapon related skills; Attack Speed, Recoil, Reload Speed, because they were easier to extract information for. I have also added a preliminary list of skills with CC effects based on whether there was yellow text present in the description.
This project has come a long way from its original conception, and a lot of the changes have been outlined in later posts to this thread making the original post a little outdated. I will try and cross out anything that is no longer relevant as time goes on, but I want to keep it there so people can see how this project as changed over time.
I’ll try and add other skills once I have more information. There is a lot of stuff to sift through, and that’s going to take some time to fill out 
This could be useful for quickly deciding which gear skills you should focus on and/or flawed skills you should avoid/choose when setting up a character specific Loadout. I will go into more detail after the break.
This project how now been renamed to “Character Builder” to better reflect its purpose which has gradually changed over time.
Below is a preview.
Version 1
Version 2
Version 3
Version 4
Screenshots go out of date! This is what it looks like now
The key thing to understand with character skills in Battleborn is that the game is built upon a FPS engine (with some stuff likely carried over from Borderlands 2) using a heavily customised version of the Unreal Engine 3. This means that everything has to more or less conform to the logic of a shooter, and the underlying system, this includes non-standard weapons and skills.
Miko’s Kunai for instance are treated as an Assault Rifle internally, which is perhaps why if you’ve ever thrown Miko’s Kunai before you’ll probably note how accurate (and awesome) they feel.
One belief amongst the community is that Attack Damage gear will actually boost Miko’s healing. I can neither confirm or debunk this theory from looking at the data, but an interesting thing to understand is that all forms of healing (health and shields) are actually a form of damage. In a sense, when you heal a teammate, you are actually dealing damage to them but you are simply adding the damage instead of subtracting it away. Due to this fact, it is entirely possible that increasing your Attack Damage may increase the healing given, however because there is already a Heal Power skill, I would think that this would be unintended behaviour if you can actually stack multiple skills into healing like that.
Life Steal is another character skill that players often ask about. Most of the in-game descriptions imply you get back a portion of the damage you deal as health (or shield), so the more damage you deal, the more Life Steal you will receive, meaning increasing Attack Damage and/or Skill Damage is probably going to be best way to increase your Life Steal abilities.
In regards to Healing Received, I’m not too sure if Life Steal is increased by it but I have always assumed it has to be from an external source in order for Healing Received to have an effect. External sources should include healing objects/traps that you yourself have set up. If someone has actually properly tested this and knows for certain how it works, please let me know.
Turns out Heal Power affects all forms of healing, this includes self healing (this also includes shields). Healing Received does not buff Life Steal effects. Thank the Gearbox devs for elaborating on this!
Some gear skills will only affect characters that have weapons or skills that rely on them. This means if a character doesn’t use a particular gear skill, you could use flawed gear with it as a de-buff without worrying about the negative effect and benefit from the cheaper activation cost.
Attack Speed can be confusing since the gear type icon is of a sword, but remembering that Battleborn is a FPS, Attack Speed and Fire Rate are actually the same thing. This would only affect the speed in which you fire a weapon though, not increase the speed of projectiles fired. So like Benedict’s rockets will only travel faster if you use Speedy Delivery (+25% Rocket Speed).
You might note that not all of the weapons listed have an Attack Speed, this is because my data mining efforts couldn’t find a Fire Rate value on those weapons. This doesn’t necessarily mean they aren’t affected by Attack Speed (it could be using a default value that is set elsewhere), I just haven’t gone through and checked them all yet. 
Recoil Reduction is another stat that I think requires a weapon/skill with some form of Kickback in order to have an effect. So if your character doesn’t have anything with a Kickback, this is another stat you could have on flawed gear without any negative effects. Kickback can be in the be in 3 directions, Up, Left and Right and it varies from weapon to weapon.
Reload Speed, is a stat that requires the character to have a weapon which has some form of a reload. Not all character weapons require reloading so if you have a piece of flawed gear with a Reload Speed de-buff, it will have no effect (this is my favourite type of flawed gear, since only a few characters will actually use Reload Speed). Some of the characters I have listed as having a Reload Speed may not appear as obvious as others such as Montana who has infinite Minigun ammo, but the weapon data I collected indicates he must use Reload Speed as perhaps either part of his Spin Up sequence or maybe Overheat cooldown (I’m not certain which it is, all I know is he is being affected by Reload Speed in some way).
Update: It looks like weapons that overheat (such as Montana’s Minigun) benefit from Reload Speed buffs when on cooldown. So more Reload Speed = Less time it takes for a weapon to cooldown from overheat. Thanks to @Amionellion and @Zarnias for testing this.
Eldrid by default don’t come with a shield, meaning you can use flawed gear with a negative Shield stat without losing anything, however certain Eldrid skills can create a temporary shield and you can also give an Eldrid an artificial shield (albeit smaller than most) through your loadout as this can give them an extra layer of protection that could help preserve their health rather than having no shield at all.
CC Duration, stands for Crowd Control Duration, which is any effect with a disabling effect.
The full list of CC effects include (some of these may share the same CC effect, but the terminology differs between characters):
- Stun
- Completely stops targets
- Blind
- Prevents targets from seeing
- Silence
- Prevents targets from using Skills
- Slow
- Reduces target Movement Speed
- Wound (Galilea)
- Prevents targets from healing
- Pull (Forced Move)
- Pulls targets towards a focal point
- Push / Knock Back (ie Montana’s Lumberjack Dash)
- Send targets backward, if they hit something, it will trigger a Stun effect preventing any movement for a brief amount of time
- Knock Up (ie Wrath’s Rath’s Catalytic Smash)
- Sends targets into the air preventing any movement until they land
- Reduced Enemy Damage (Attikus/El Dragon)
- Decreases outgoing damage from affected targets
- Damage Amp (Galilea/El Dragon)
- Increases incoming damage on affected targets
- Reveal
- Disables active Stealth abilities
- Target/Mark (Reyna/Marquis/Benedict)
- Targets are marked for a brief amount of time causing unique effects based on the character used
Any target that is afflicted by a CC effect is considered weakened (characters such as Deande can deal bonus damage on weakened targets). Having played a bit with Deande, I’m not totally convinced her Weaken Effect is the same as similar “Reduced Damage” effects from other characters. I assume you will only get the bonus damage from “The Culling” (R7) provided you first hit your target with a Burst Dash. If anyone can knows otherwise, please let me know!
A quick way to know if a particular skill has a CC effect is to see if there is any bright yellow text in the skill description.
The ones listed on the chart are for seeing what abilities create a CC effect, since CC Duration gear only affects incoming CC.
Some of the skills I have listed with CC might not initiate the effect but might do something if one is active (such as Mellka’s Venom having a Slag-like affect on many of her skills). CC Duration on gear is only for incoming CC effects, so it is more useful if you want counteract other characters you might come up against that have a lot of CC skills.
Ok first pass on characters is done. I think the further I went on the more things I started adding in an attempt to build up a quick reference chart for characters.
Here’s a list of skills that are shared between characters but don’t really have a gear type that boosts them (yet).
- Damage over Time (not really a type, but its important to display since the way damage is being applied can have a big impact on how much of that damage is actually registered)
- Overshield (temporary instant shield boost like the ones you pick up in PvE)
- Shield Steal (functions exactly the same as Life Steal except for shields instead)
- Damage Amp (think BL2 Bee Shield)
- Area of Effect (ie radius)
- Shield Damage (bonus shield damage)
- Damage Reflect (think BL2 Sham Shield)
- Accuracy (Weapon Accuracy)
- Attack Range
- Magazine Size
- Instant Health (like you just picked up a health item as opposed to Health Regen)
- Projectile Speed
Both Kleese and Reyna have the unique quirk of being shield healers, which I haven’t settled on how I’m going to represent it on the chart. For now they are simply Heal Power (Shield) and Healing Received (Shield).
Not all skills/augmentations directly affect the user, so I’ve also added in a mention if the target is say another player or a turret setup by a skill (ie Toby’s Barrier, Kleese’s Rifts, Ambra’s Sunspots).
- Shared - Yourself and all nearby teammates will benefit from the effect
- Given - The effect is given to a teammate or minion
- Turret - The effect is applied to turrets setup by your skills
- Reset - Triggers a stat reset (ie Cooldown Time, Shield Delay, Shield Recharge)
I’m now looking at adding another layer of detail to this chart (without hopefully overcomplicating it too much) so you can make further observations about character augmentations without needing to read the descriptions. I realized that the Gear Rarity system could be used here to describe how a skill is changing a stat.
- Common (White) - Flat one time effect (ie Reveal, Damage over Time)
- Uncommon (Green) - The effect requires certain conditions to be met and has a duration in which the effect will last (in much the same way Uncommon gear skills behave)
- Rare (Blue) - The effect still requires certain conditions but doesn’t have a duration (ie effects that do more damage based on proximity to the target).
- Epic (Purple) - A flat permanent boost to a stat that remains until the end of the match.
- Legendary (Orange) - A recommended gear type that would boost the given effect.
The last one will mostly be used in the Skills and Weapons sections, but if an Augmentation adds a new effect not previously available (ie adding damage to an attack that doesn’t do damage by default), I’ll add a recommendation there.
Not all damage dealing attacks will deal out damage the same way. Understanding these differences can make a big difference in how you approach with certain attacks.
- Instant Hit Damage
- Deals “instant” damage when an attack lands on a target (ie Melee or Projectile Damage)
- Damage over Time (DoT)
- Damage is applied over a period of time. It comes in 2 forms, Bleed Damage, and AoE Damage
- Bleed Damage (DoT)
- Once a target has been hit, they take a set amount of damage over a period of time regardless of their distance from the original point of contact
- Area of Effect Damage / Pulse Damage (DoT)
- Damage is applied in reoccurring pulses over a set period of time and any target within the Area of Effect for each pulse will take damage. If a target moves out of the Area of Effect, they can effectively reduce the amount of damage they will take from an attack.
With AoE Damage in particular, you want to try and keep your target(s) within the AoE for as long as possible to maximize damage output. Applying other CC effects and/or timing your attacks with other teammates are just some ways you can achieve this.
This is still a work in progress and was mainly put together programmatically (26 Skills/Augmentations + 1-3 Weapons x 25 Characters is a lot to manually go through), so the next step will be refining this chart and adding stuff that is missing, fixing errors etc. I am by no means an expert when it comes to this stuff. 
I am slowly working through each character and manually adding all relevant skills into this chart.
Feel free to add your thoughts to this discussion, especially if you’ve heard of any rumoured gear stats affecting character skills or if you disagree with some of the assumptions I have made here. The more information I have the more accurate and useful this chart will be
Good luck on it!
. The drawback of trying to do stuff programmatically…
