A multimeter isn't a necessity - with a little ingenuity you can test for more than just voltage with a simple 12V test lamp. If you don't have a test lamp but do have a soldering iron, then you could remove one of the little interior courtesy lamp bulbs and solder a couple of leads to it, then wrap insulating tape around the exposed metal to keep it from shorting to anything and you're ready to go - you just need something to indicate the presence or absence of 12V.
It's by no means unusual to find unused connectors in cars. Instead of making a different wiring harness for basic models and another for mid-range and another for top-of-the-range, it's a whole lot cheaper to just make a single harness to cover all variants of a particular model, and then simply leave plugs disconnected if they're not required, so that connector may not be anything to worry about. You could trace it back and see what it connects to, and take a note of its wire colour and then find it on the wiring diagram and identify its purpose from that.
As for the second photo, if that's an insulated wire with the end bared and then secured to the bodywork by a screw then that would be a ground connection for something, and it sure doesn't look too healthy, so I'd remove it and clean it up and reconnect it. It would be a good idea to smear it with a protective coating of petroleum jelly (Vaseline) before reassembling it, and that would help prevent it from corroding again in future.
Edit: I was looking more closely at the second photo, and there's a bit of that wire that doesn't look quite right, which I've put a circle around in the photo below. If that's just a knot in the wire or if there's something stuck to it then it's probably ok if the insulation isn't broken, but if the insulation is damaged and you can see bare wire then the copper is likely to have corroded internally, in which case that section of wire should be replaced.
