You need to do several quests for the locals and then do the "identify the skooma dealer" quest before the jarl will offer the house.
No, just five quests.
You also have to give a health potion to Wujeeta (?) and then ask her about her dealer and the talk to the Jarl and everything else should become clear.
Dont think TG counts, since its technically the "underground" and not actually a part of the city. I would suggest just doing the quests for people to ensure it all goes properly.