Simple question. If I do not, how do I do that?
If I do, can I double-cross Ulfric and join the Imperials?

no.

you can speak to the steward (Jorleif stays in that job) about getting a house and he'll give you quests.

Be careful though... Blood on the Ice is required to get the house. DON'T BUY THE HOUSE BEFORE YOU HAVE COMPLETED THE QUEST CHAIN unless you have researched the chain on a wiki or here in the forums... it can be really frustrating!

Warning and spoiler alert here... search the forums for Blood on the Ice... it's a buggy chain and if you go through it carefully you get the best house in the game.

Also... because I'm giving spoilers anyway... when you DO get the house, ask Jorleif to clean it before you buy anything else. THEN go to the house, make sure it's clean and there's nothing there that shouldn't be (like proof you found from doing Blood on the Ice) and save your game before you decorate. Sometimes the house bugs and leaves some of that "proof" behind and you can't get rid of it. LOL

it is still totally worth the effort on this house. I cannot repeat often enough how amazing this house is compared to some of the other more expensive ones. Kinda rowdy and uneducated neighbors, and still absolutely an adventurers house.

per wiki: "Hjerim is purchasable having sided with either The Empire or Stormcloaks. Once allied with the Stormcloaks, the http://www.uesp.net/wiki/Skyrim:Fort_Neugrad http://www.uesp.net/wiki/Skyrim:Rescue_from_Fort_Neugrad must be completed before http://www.uesp.net/wiki/Skyrim:Jorleif will make the house available for sale. However, once Falkreath Hold has been given to the Imperial Legion as part of http://www.uesp.net/wiki/Skyrim:Season_Unending, it may be necessary to liberate Hjaalmarch before Jorleif will make this house available. If http://www.uesp.net/wiki/Skyrim:Imperial_Legion is chosen as an ally, then the defeat of the http://www.uesp.net/wiki/Skyrim:Stormcloaks is required before it is available to buy."

I have never had issues with the quest chain Blood on the Ice! LOL The only issue that I experienced is after the quest is done and I've bought the house, I never saw the "clean the house" option that you are supposed to be able to purchase from Jorleif. I just used the console to clean up the bloody mess lol

But yea, you can buy the house joining either war faction... (to go back on topic)