At the beginning of the quest phase, I am engaged with an enemy. I commit Halbarad to the quest. Then I reveal "Surprising Speed" from the encounter deck, and thus return my engaged enemy to the staging area. I am now not engaged with any enemy. Is Halbarad still committed to the quest? If so, does he now exhaust?

Thanks for your answer!

Halbarad is still committed to the quest and does not need to exhaust. the requirement to exhaust or not to commit to the quest applies when you commit at the start of the quest phase, then it's done and cannot be changed (except by things which specifically say to remove characters from the quest).

Committing to a quest is always done before you reveal any encounter cards. Just look at the state of the game now. There is an enemy engaged with you? Now Halbarad doesnt have to exhaust for committing while other characters probably have to exhaust.You also have to apply any forced effects and may apply reaction effects already in game now. When all is set and done you start staging and reveal encounter cards one by one.

Just treat Committing to a quest as a single step you have to finish completely before you go to the next step Staging.