How does regeneration work if damage does not go into the stack anymore?

For example, I control a River Boa and block my opponent's attacking Dwarven Vigilantes. If I regenerate my boa, does it still deal damage to the dwarf?

And how exactly, step-by-step regeneration works during the combat phase? When do I need to regenerate a creature? At the beginning of the damage step?

Played a draft another day and short circuited my brain when I tried to apply regeneration to the "new" rules.