Sounds like a great deal!
Yes, '09+ has the 6-speed. I believe the exception is the PPV got it in 2010+. Factory Bluetooth isn't all that special. It's for phone calls only. My personal recommendation is to get an aftermarket HU and you'll have tons more features, phone and music streaming via Bluetooth (or Wi-Fi if you get a fancy enough HU) and better sound. Other than the better trans, about the only real mechanical issues that were improved upon was the AFM components and the driver's side rocker cover.
The gas cap may have been replaced with a non-FF OEM or aftermarket. I believe the only difference is the color. The only way to tell if it's FF by looking at the engine is to ID the injectors. You can double-check your build specs by looking at the RPO label on the inside of the glove box lid. Look for "LMG".
Your '07 came with the "first and worst" iteration of AFM. There's a chance the parts were updated, though. They improved the VLOM and lifter designs to greatly increase reliability. Since yours is low-mileage, you're in a great position to disable it and let it ride. My personal recommendation is to have it tuned out. You can get someone local to do this and it'd likely be at least $150, which is still cheaper than the plug-in module options. Or, you can send the PCM to
this guy and have it done for about $70 or less, depending on shipping. If you wanna splurge, get a full custom tune by someone who specializes in these (such as Black Bear) and you'll get all kinds of improvements from better drivability from more responsive throttle and shifting characteristics to increased trans life. Above all, use quality oil and change it regularly, before it gets too nasty. My experience shows that 5,000 miles is a good OCI for when the oil starts to get dark. Use of a quality full synthetic oil and good filter is paramount.
As this is a common problem, you're suspicions may be correct. Nobody here knows your husband's mechanical aptitude. But, I'd rate replacing the door latch as a "3" on a 1-10 scale. Somewhere above an oil change but a under a brake job, if that tells you anything.
Batteries often show their faults when the weather changes, especially from warm to cold. Did it fail during a cold snap in your area? It may charge up and hold enough voltage to start and run it and seem "okay", but it could have an internal fault that the fancy electronic tester could see. Of course, there could be other possibilities. You can do a simple test with a DVOM (digital volt/ohm meter) to see if there's a steady drain and on what circuit. You set the meter up to read amps, disconnect the negative cable from the battery, put one of the meter's leads on the battery post and the other lead on the cable. Take note of the amps (should be milliamps). If they're high, remove one fuse at a time until the measured draw plummets. This will be the circuit that has the drain it. Be aware that, out of the many computers in this thing, some are still active long after the key is turned off and removed and the doors are shut. Some may reawaken from the disconnect and reconnecting of the battery, even if through the DVOM.
Could be tires, trans, front or rear differential, hubs, etc. Make sure it's not in 4WD (neither 4HI nor 4LO).