For any one who has found this thread and is still stuck, I solved this by:
Getting a tyre lever inside the fuel cap and gently forcing it open a bit whilst repeatedly pressing unlock on the key fob.
Once its open you can look into preventing it from happening again. I ended up taking the whole fuel cap section out and duct taping a bit of metal in the way ...
A new starter will spin the motor faster using less current saving wear and tear on the alternator.
Alternator isn't working till the engine is up and running. It is the battery which powers the starter motor, so no wear and tear on the alternator
Look at the lower left-hand corner. It says "VW AG" which indicates this is a VW OEM battery. The "2509" says, I assume that it was made in the 25th week of 2009 which is in line with your 2010 model year.
As to whether this is the original battery in your car, it's difficult to say with certainty but this sure looks like an original ...