A player starts a chain and play an action. Is that same player can play a reaction before the others ?

Yes — you keep priority after playing your Action, so you can play a Reaction onto that same chain before opponents get a chance to respond.

Finalizing a card doesn't pass priority, and the controller of the newest chain item receives priority, so the first player with priority is the one who started the chain and may add another item to it (Rule 338.1.a.5). That only holds for another Reaction — you can't add a second Action to the same chain (only Actions can start a chain; once one exists, only Reactions can be added, Rule 309.1.a). Priority passes only when you choose to pass it.