Personally, I would convert them to Crown Gems that you can use to buy specific items from crates ... but to answer your question:

Weapon poisons have a 10% chance to proc from eligible weapon abilities. So that includes light and heavy attacks, and the direct damage portion of most (I think?) weapon skills. So class skills won't proc it, and they won't proc from the backbar. Otherwise, not sure why they wouldn't. They should work for bows. If you just light attack an enemy 10-15 times, you should see a proc.

Equipped poisons have a chance to work on weapon attacks or light/heavy attacks. Note that you have to equip the poison on both bars if you want to use the poison on both bars -- otherwise it only can happen on one bar. You may need to split the stack into two stacks just to put them on both.

I'm not sure if the poison cooldown competes with enchant cooldown, I don't usually mix them.

Not all skills count, but generally you have a 20% chance to apply poison with direct damage attacks or weapon skill line attacks. Many AOE attacks that deal damage repeatedly can also work.

The crown poisons can work, you might hear a little hissing sound with a tiny animation on enemies hit with the poison.

On the subject of *should* you use the poisons from crates ... I'd say no. Convert them to gems whenever possible.

You can craft or buy equivalent poisons with Alchemy/in-game gold. The crown version offers nothing except they're usable by any level.