does AP work with posion? Does it work with Shake off?

If it deals one or more damage, it'll work with Armor Piercing. Regular attacks, RCE, CCE, Posion, EE, Shake Off, etc. all deal damage.

If AP works with EE, does that mean that multiple target EE's stack, so that potentially a 3-target EE attacker could do 3 to a single target, if that target shared a splash square with three adjacent, legal targets? And then even more, with a critical hit?

No. EE is a single attack, so AP will only work once for 1 damage. Doesn't matter if there is 1 target or 20 targets (hmmm... who would have 20 targets? Flash with a gun?). Think of it this way: for an EE attack on 3 targets, how many times do you roll the dice? Once, and you compare that one attack to all the DVs. So AP only works one time, not three.

Hope that helps.