...are characters with which they are engaged able to Refresh in the next roundor do they remain exhausted, committed to the combat? If they are still engaged and exhausted, this would mean that they cannot quest. So this is an important clarification for me.

Monsters don't become engaged with characters. They become engaged with players. Which means characters are free to do what they want once they ready.

Thank you for the reply. So you are saying that Ally's and Hero's both Refresh even if combat is still occurring from one round to the next?

Yes. That is correct.

Otherwise, something like Ungoliant's Spawn (9 hit points, 2 defence) would be even more foul - you'd either need to hit it with 11 attack in one round, or have most of your characters tied up for the duration.

Yes, you kind of think of it as your whole party of heroes and allies is able to switch off who is doing what with ease, and cover for each other, so they allow themselves time to recover each round. Heck, maybe later on there will be an enemy that focuses on 1 on 1 battle and does not allow your character to do anything else until defeated (or if they kill a character, another must be chosen or some ill effect happens.) That would be an interesting idea. They really could still go in so many directions with new cards.