Gen 2 key fob malfunction?

My 2008 has needed a couple of jump starts recently. Also, now the car is stuck in neutral, not starting, and won't release the key fob. Lights work but fob stuck and no start. To make things worse, the car is in Waterville, Maine which is 1500 miles from me. Any ideas?

Update: The key fob came out and I've been informed that the lights don't work. Other information is correct. I'm going to have my daughter check the battery at this point.

 

Battery is fairly new...maybe 5 mos? AutoZone or something like that AGM for around 240 bucks. It's in Maine so I know that's cold up there. Also, do not know what the guarantee on it was. Apparently, dash lights did come up while it was being jumped or attempted to be jumped. No other signs of life tho.

 

I'm leaning toward the idea that the battery is low but I'll have her get the battery tested. There was a recent similar failure but a jump start was successful in that case. Here, the jump start had no effect.

 

I will start with battery testing. Assuming the battery is good (it's only about 6 months old but still could be defective) do you have an opinion as to what parts would be most likely damaged by improper jump starting? If I repair this, I might take a bunch of parts with me just to be sure I have what I need because of how far the car is from home. I would get these parts mostly from part out cars or the junkyard and then sell the parts not needed later.

 

Thanks. I've heard the 08 and 09 inverters are better than the earlier gen 2 but do not know whether that means they're better protected against frying.