Most (all?) of Zane’s cryo-based skills are down the Under Cover (green) tree, so you may as well rock a Barrier as part of your build, as you’ll have access to all the related augments.
Specifically, points in Brain Freeze makes it easy to freeze enemies regardless of using a cryo weapon or no (critting with cryo weapons makes freezing that much faster though), and Calm Cool Collected turns these freezes into huge bonuses for Zane (in order: shield fill, health fill, and then full action skill cooldown and reset).
Because you’re hiding behind the Barrier and not taking damage, your shields and health will be full, so a freeze will refill your Barrier meter (and whatever else you’re rocking - dealer’s choice here). If you can make a stand behind the Barrier, shoot and freeze one enemy, damage them a bit more, and switch to another so you can keep the freezes up (there’s a little cooldown between freezing the same enemy a second time). You can keep your Barrier (and whatever other action skill you like) up indefinitely this way, short of them being destroyed.
For weapons, it’s still almost up to you… there are plenty of good cryo weapons capable of freezing enemies to get these skills lit… it might be better to tell us what your preferred playstyle is than throw a list of several dozen guns that are good at cryo.