A player may choose any of the four core methods they'd like for each of their characters. They don’t need to use the same method for every character.
See Prime Requisites.
Characters add their CON-based HP adjustment to each hit die, not level. That means that monks and rangers begin the game with twice their HP adjustment.
CON gives a bonus to hit die rolls. Therefore, HP are not retroactively adjusted if the HP adjustment from CON changes. The new CON score does affect future HP rolls, however.1)
A character’s original constitution score (after race is applied, but before aging is applied) is the maximum amount of times a character can be raised from the dead or resurrected. Beyond this number, raising is only possible through wishes. Since a character loses 1 CON every time they're raised, without some magical means to recover lost CON it's more likely that the character fails a Resurrection check before they hit their maximum raises.
We have Foundry macros for Methods I-IV.