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

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.