If I have a unit at a battlefield and trigger Temporal Breach on the opponent's turn, does it trigger another combat/conq? Do I lose control during the spell?

No to both — Temporal Breach on your own unit on the opponent's turn does not trigger a new combat/showdown, and you do not lose control during the process.

Here's why:

Control is preserved. During Temporal Breach's resolution the game is in a Closed State (the Chain is active), so Rule 187.4.c's loss-of-control check doesn't occur. After the spell resolves, your unit is pending on the Chain (being played back) — and per the Baited Hook FAQ clarification, a pending unit on the Chain prevents loss of control. By the time the unit finalizes and enters the battlefield, you still control that battlefield.

No combat or conquer. Since you never lost control of the battlefield, your returning unit does not apply Contested status (Rule 187.3.a.1 — Contested only triggers when a unit's controller does not already control that battlefield). Without Contested, there's no Showdown and no combat. And since you haven't lost and regained control, there's no conquer either.