Re: zero Question
Yes, it all depends on points of contact or lack there of from the stock to barrel/action and just about all stocks are different. Even the same stock can have different effects on a rifle when the action screws are not torqued to the same values each time it is removed and replaced.
You might not see a big change going from one well bedded stock to another that is bedded properly but I'd imagine that no matter what, there would always be some change. The lesser the stock or bedding or lack of bedding, the greater the change in POI.
You also might have a small change if going from one free float Manner's type mini chassis system to another chassis style system. At the same time if you went from a cheap factory wooden stock to another cheap aftermarket composite stock you might have a small change, but... they might both shoot like shit haha.
If you don't already have one, get a good stock and have it bedded and properly torqued to spec or get a stock with a chassis system and buy a torque wrench, this will save time and money in the long run in my opinion. Good luck.