Can You become the Thane of Riften if a NPC dies before you complete a quest that's required?

I know it's not the best way to get there, but you can revive the NPC using the console.

Go into the console, use the mouse to click on his body, then type in resurrect and hit enter.

(There is a Guardian Stone outside of Whiterun that teaches you to raise the dead. But I believe they turn into zombies instead, which is not what you want here.)

Nope, The Skooma Trade is the opening quest to Becoming a Thane of Riften, so you have to finish it before being able to start the quests leading to being Thane. Resurrecting her will still consider her dead, whether you use magic or the console.

From then, your choices are:

• Pick a previous save, before the dragon
• If on computer, you can force start the quest using the console (~ key), and the command setstage FreeformRiftenThane 10 ; save before doing it though, just in case

Yes, if the Empire takes control of the Rift you will become thane fairly easily. It can happen either via the civil war quests or the main quest, during the peace council.

If you are playing on a PC

SetStage FreeForm12 10 - This is where you are if Wujeeta asked for a healing potion.

Assuming Wujeeta is dead, then

SetStage FreeForm12 20 - Not sure what it does, but I did it on the way to the next step. Can't hurt.

SetStage FreeForm12 25 - Gets you the reward. Enter Riften proper, then go Back to the Docks Area. Wujeeta should be wandering around now and you can continue on with the quest line.