No it won't kill the tree. But you might need to water more often or prune the top. Basically the top of the tree may have outgrown what the potting media can support. Properly cared for it won't die. Peaches can stand a lot of pruning. So don't be afraid to do that if it needs more watering than you can manage. I water mine once a day in summer.

No, I have that happened, I wait when they are dormant, during winter, then pull out and try to save as much from the root and move to bigger container, never had lost one this way so far.