It seems that when I finish side dungeons that aren't marked as giving out an aspect, my number of side dungeons done don't increase, and neither my renown.
Is this normal ?
There doesn't seem to be as many dungeons with aspects as the max number listed.
Aspect rewards are per-realm. So if another of your characters has already finished that dungeon and unlocked the aspect, it will not show any aspect reward for a different character.
As for not getting the renown progress, is it possible that you've already done that dungeon with the current character?
Hmm. Which dungeons would those be? As far as I'm aware, all the main dungeons give an aspect as a reward the first time you complete them. These will also count towards the renown goal.