Do side quests count as the current quest if CFA is revealed while is chosen as the current quest?

Hmmm it looks that way which means you can much more easily get rid of Cold from Angmar (from questing at least) if a side quest is in play. I don't have the lost realm just yet however so maybe check what the rulebook that comes with it has to say about side quests and see if that info can answer your question as I am not sure on this.

Do side quests count as the current quest if CFA is revealed while is chosen as the current quest?

Yes, certainly they do: it's very clear in the rulebook:

"At the beginning of each quest phase, if there are one or more side quests in the staging area, the first player may choose one to be the “current quest” until the end of the phase instead of the quest card that is currently active via the quest deck."

A chosen side quest is the "current quest" until the end of the quest phase and will therefore gain Cold from Angmar.