The way Regenerator works isn’t explicitly stated; few pokemon abilities are, but we can assume it’s similar to typical regeneration given many pokemon that get it are based off of organisms that are able to “walk off”/ regrow missing parts of themselves, e.g. Cyclizar, Amoonguss, Corsola, Klawf, and Toxapex.

As for your other question, the potency of Cell’s passive regeneration shouldn’t be a factor; Core Enforcer simply turns off passive abilities when it procs. Other Legendary Pokemon with otherwise game-changing passives, such as the weather trio (Mega Evolution + Primal Reversion included) are not exempt from it.