If the overlord won the previous quest, can the overlord choose to play a rumor quest that is in play?

Generally speaking, yes he can. If the overlord has the option of choosing the next quest, a rumor in play is going to add to the list of available quests.

Yep, if the casmpaign allows for the overlord to choose the next quest, available rumor quests are not special; they are eligible just like other campaign quests.

But a competitive OL would never want to choose a rumor quest, right?

Maybe if ol is confident to win and really wants a specific reward (at cost of search card gold). I mean, it's probably more fun too, right?

I always play them because it's more fun, even if my heroes may end up more powerful than I'd like them 😛

It's still can be a good option to play them to gain some usefull threat tokens for plot deck, from 1 to 6 more if the quest is played and rewards if not played when tranditionning to act 2.

Enough avantage to me to be played as overlord (except maybe some of Lair of the wyrm since relics are far too powerfull for the heroes)