Just some of my ideas:

2) Catching a train would be the easiest option in my opinion. At FRA there are two train stations:
1. regional train station: local trains (RB, RE, S-Bahn), should cost you 5-6 € and 20-25min to get into town (tickets need to be bought before getting on the train)
2. "Fernbahnhof" / long distance train station: I would recommend taking an ICE train to FRA central station, takes you 9 minutes into town (11,50 €) (tickets can be purchased on the train using cash or MC/VISA)
Taxi would cost you about 30 Euros and would certainly take longer (between 7 - 9 a.m. there might also be some heavy traffic on the motorways)

3) Try to catch some sleep (3 - 4 hours minimum) on the plane and try to stay up all day till at least 8 p.m. (At least thatīs what I always do and up to now I have never suffered from any jetlag)

4) I must admit I donīt know much about Frankfurt City itself, but there are many places / cities nearby you might be interested in: plenty of little vineyards, castles, the river Rhine, Heidelberg (max. 1h on the train)
Donīt know how long you will be staying and how much time you have to travel around, but other options would include: going to Cologne (1h on the ICE) or Munich (3,5h on the ICE), going across to France, catching a plane to Berlin etc. Basically Germany isnīt that large so depending on your interest there are plenty of options that are doable for a one or two day trip

5) Not sure about AA, I just know that when I last arrived at 7.00 a.m. at FRA (travelling in J on AC) I was told there was no fast lane I could use, but customs generally doesnīt tend to be that bad at FRA (at least from my personal experience, Iīve never waited more than 15min) For more details there also is a section on FRA in the Germany Master thread

Hope that was any help to you. Enjoy your time in Germany!