Can I conquer the same bf twice in a turn if I conquered it, then the opponent conquered the same bf and then I conquered again?

No. Once you've scored a battlefield (by conquering it), you cannot conquer it again in the same turn — even if the opponent takes it and you take it back.

Why: Each player may only score each battlefield once per turn, by either method (Hold or Conquer) — Rule 465. Your first conquer already used up your scoring opportunity for that battlefield this turn. When you later regain control, you take control but do not score again, so it doesn't count as a conquer and won't trigger conquer abilities.

Your opponent can score it when they take it, since they're a different player and haven't scored it yet (FAQ 244). But your second reclaim is just a control change, not a conquer.