2000 Mercury Marquis 2 Problems

2000 Mercury Marquis

Hello,
I have two problems that I'd like help fixing. I have a 2000 Grand Marquis.

1) My lighter doesn't work. I try charging my GPS and it doesn't work. How do I troubleshoot this and work to fix this?
2) For some reason, my button unlocker doesn't work anymore. The one that you can enter a number combo in to unlock the car. It was working fine and now all in a sudden it doesn't work. How do I troubleshoot this and make it work?

On side of that, I have one other question out of curiousity:
3) On the passenger side, under the glove box, there is something similar to a lighter outlet, what is this? I thought it was another one at first, but now I'm thinking otherwise. Any ideas?

Any help is greatly appreciated.

Thanks.