The thing with the red and green wires looks like a relay of some kind. The red wire is definitely a ground, it goes to the frame. Red is not normally a ground color, they are usually green or brown. Red should be hot. But you can tell from the crimp on connector on that red wire that it is a home spun job.
The part with the rotted hoses appears to be the vacuum fuel shut off valve, but I can't tell for sure. If it is, it will have a hose from the fuel tank, one to the carb, and one to a vacuum source somewhere near the carb.
Adding new acid can rejuvenate an old battery, but I have never seen it last more than a few weeks. Chances are you will still need a new battery soon. But $60 seems like about twice what it should cost. I got a new sealed battery for my XT225 for $50 that is still good after almost 3 years, from Batteries Plus.