I’ve only looked at weapons and Moze’s Mind Sweeper class mod micro grenades with any depth however the 5 minutes of throwing grenades at the target dummy suggests all the Mind Sweeper calcs apply to general grenades.
For Guns, AOE and Splash appear to be the same thing. I’m yet to see a weapon that is affected by 1 but not the other. They are also added in the same modifier in all damage calculations I’ve done. I’ve not found any gun affected by grenade damage.
The Mind Sweeper grenades are affected by all 3. The AOE/Splash are again combined and then multiplied by the grenade boost.
So in answer to your questions
Grenades are affected.
AoE appears to include splash. I believe the relation is probably that being splash implies that it’s also AoE. So yes Aoe will affect splash and grenades.
Weapons and Grenades get splash. I believe certain artifacts do too.