I cast a spell then hold priority and use a reaction card, then I pass. Now my opponent responds by using a reaction card. After he used it, does he still have priority as well to use another reaction card before passing back to me?

Yes, after your opponent plays their Reaction card, they retain priority and can play as many additional Reaction cards as they want before passing back to you.

Here's the sequence:

1. You cast a spell → you get priority
2. You hold priority and play a Reaction → you still have priority
3. You can play more Reactions, or pass
4. You pass → opponent gets priority
5. Opponent plays a Reaction → opponent still has priority
6. Opponent can play more Reactions, or pass back to you

Priority works the same for both sides: each time you add something to the chain, you keep priority and can keep adding more. You only lose it when you explicitly choose to pass. (FAQ #5569, #1066, #4696 — Rule 338.1)

So no, playing one Reaction doesn't force them to pass. They can chain multiple Reactions together before sending priority back your way.