Congrats on the Jetta... Looks clean from what I've seen so far...
I'm not sure about the Wires from the battery... All I can think would be Positive, Ground, and Monsoon; but I never really put much thought into it. Hopefully someone else will chime in on this, but I'll see what I can find for ya.
For the alarm... From what I understand, if you want to have it go off when something hits the car, you would need to add a shock sensor to the system. Most people just opt for adding a 3-rd party alarm (Not VW) when going this route.
Headlights... I had a similar problem with mine. Are yours yellowed and clouded at all? I cleaned mine up with some headlight restorer (plastic polish basically) and it helped a lot. Currently, I am looking to switch out my stock headlights for some "E-Code" ones. They have the European-spec light beam pattern, and from what I have seen they light up the road better.
This box on top of the battery holds some fuses and I would believe would be the grounds as stated above.