1# In general how do miners pick a nonce? Obviously if everyone started at 1 only the fastest miners would win. Do they just pick a random very large number and start incrementing from there? How often do they update their block (if you do it with every single new transaction the instant it comes in it seems you might take a performance hit on your hash rate).

2# Once a miner solves a block, how can they prove it was them who solved it? If Bob tells Alice the solution, what stops Alice from taking the solution and quickly trying to tell more people quicker it was her who solved it?

edit: Removed two questions above as the miners use their own bitcoin address to create the header hash which is used to solve the block.

3# Seems the single largest miner is about 20% right now. Let's say I want to know how safe I am after 5 validations in the worst case scenario. Does that mean .2^5= 0.00032 (aka 0.032% or 1 in 3125) is the chances that a block with 5 validations will later fork if that 20% decided to try to cheat the system at that moment?