Will bombing the donation machine re-lock already unlocked items

Edmund explains it here, basically it slowly upgrades the store, sort of leveling it up to have more items, etc.

Lazarus starts with different items than Isaac does. Notably, he starts with an item that gives him an extra life (hence the name Lazarus).

He also seems to start with a random pill... but unlike Isaac, Lazarus doesn't start with any Bombs.

Edit: I haven't gotten far enough to check this, but in the original game, Isaac would start with The Dice if you'd unlocked them... Lazarus likely doesn't.