You got a warning so no worries. Sometimes an officer will follow you and claim he was trying to catch up or safely pull you over...sometimes to see if you do anything else wrong or questionable in order to make a decision on possible tickets or just a warning. Either way, an officer is likely to claim it was the safety reason if questioned about it at some point..
Statute of limitations has to do with how much time elapses between the incident and when you are officially charged with something. With a traffic ticket, you are charged right there, same time and place. There's no statute of limitations issue. What you are asking is whether a state must simply forgive money due and drop the issue after a certain amount of time and the answer is no.
I don't know all the procedural aspects in AZ, but we have identity issues all the time. Speak with a local attorney about it and try to convince the court it wasn't you. Key to these matters is sometimes what identification in particular the officer was given at the time of the car stop(s).
I wouldn't expect the circumstances to matter much. The focus will be on how she actually measured your speed in the first place. You should probably speak with a local attorney about defense if you are worried about a potential conviction.
Officer "saw" something but you don't mention what if anything you were charged with. Hard to assess the need for an attorney just based on this information. It's likely you should consult with a local attorney if the charges were anything above and beyond a simple equipment (no lights) type of violation.
Generally, if your license was suspended for failure to pay something (an old ticket, judgement, child support) you need to pay it to clear yourself with the DMV. If you can't pay, perhaps there's a way to satisfy DMV that the issue has been taken care of (new child support arrangement, for ex). If you are representing yourself, your best bet is to contact DMV directly and start researching some options.