I know I have to be thane of Riften before getting the house, but for some reason, the jarl doesn't give me the option to become thane. I've done a lot of side quests and the skooma problem, but she keeps sending me in random places to kill a dragon or a giant.

Becoming thane on Rift is different from other holds. You need to deal with skooma trade in Riften and start it on your own. First help Wujeeta with her addiction and then ask where she gets her skooma. Eventually you end up dealing with local supplier and it's source. Once both are done and you've completed enough misc. quests the jarl makes you thane even without asking.

It's the other way round. You have to buy Honeyside before she'll thane you. To get the option to buy the house you have to do the http://www.uesp.net/wiki/Skyrim:Helping_Hand, http://www.uesp.net/wiki/Skyrim:The_Raid and http://www.uesp.net/wiki/Skyrim:Supply_and_Demand quests. Since you've already done those then all you need to do is help the people of Riften with some personal quests until she tells you that you've become a local hero like Mjoll. Get Balimond some fire salts, get Marise some ice wraith teeth and so on. If you've got the coin then invest in the Pawned Prawn , apothecary or blacksmith. Heck just giving one of the beggars a septim will help. Once you've done enough you should get the option to buy the house.

As long as you do 5 quests for people in the hold she'll give you the option. Just ask around for some more quests(bounties don't count)

Which shouldn't be difficult at all. Riften has the largest amount of misc quests.