There's hardly any difference between those responses. You can even say "и тебе тоже", no problem there
What's this? You explained the concept very well there, but this one kind of stands out. No one would say it like that. I suppose you wanted to say: "be careful?". If so, then it should be "Береги себя!" or "Будь осторожен" depending on the context. The former is more suited for a very caring way of saying "goodbye", the latter is what you'd say to someone who's going to do something dangerous.
Аккуратный took a stange turn in Russian, it's more about being neat (as in handwriting) or about someone who usually does things in a "neat, clean" way (like аккуратный хозяин), sometimes about a thing that's kept in such way. (аккуратная комната).
But I can guess why аккуратный can be sometimes translated as careful. Example: someone just stepped on your toes, literally. A very possible response to that can be "аккуратнее!", like be more mindful of yourself or very well be careful!Also, remember that scene from "The Fifth Element" where Bruce Willis lights up his last match to bring fire to the last elemental stone? I suppose in English the people around him would yel something like "Easy! Easy!". In Russian that can be "Аккуратнее! Аккуратнее! or Аккуратно! Аккуратно!"