No, exp should always be earned. There's no reason for anything not to count, and this would lead to abuse in PvP where players could farm infinitely without leveling up.
If you are talking about normal side quests, then I don't understand why you wouldn't want to do them. For me personally, I don't like the cheat dungeons. They stress me out which is not good for me to do, but even so, I have been trying to at least complete it once. But I know there are many who would prefer to avoid doing the cheat dungeons.
No, exp should always be earned. There's no reason for anything not to count, and this would lead to abuse in PvP where players could farm infinitely without leveling up.
You already can. You only get XP for dungeons the first time through and 1/2 XP second time. So after the 2nd run you're not getting anything except the XP from pips used.