Kat from what you posted I think there were a few of us that unfortunately felt confident your battery is toast.
Regarding what is better starting the car engine running or letting the battery take the whole load, starting the car and allowing the top to operate is always better because of the huge amp draw the battery receives. The alternator can help catch that draw down up if you let it run a few minutes.
If you can the battery tender is really helpful to prolong life. Mine is approaching 5 years 9 mos and I don't think it would have lasted that long without the tender.
Big question....is the dealer going to cover this or are you?