I'm thinking that if you have a wire shorting bad enough that you smell something burning, you'd be blowing fuses or would have already burnt the wire in two loosing a circuit completely, in which case, the smell would stop. Since it's recently had wiring work done to it, my money is on the new wiring being incorrectly routed too close to a header/exhaust manifold or pipe. You know what wires were replaced. Take 20 minutes and jack it up (use jack stands) and take a look at the wires that were recently replaced.
If you haven't gotten under the car to see if the burning smell is coming from them routing the wires incorrectly, you won't be able to claim or prove that it is something that they did wrong during the previous repair. And 9 times out of 10, if a shop can get away with saying "It's not something that we did.", they will. And then they'll happliy offer to fix the "new" problem for you ......for an addition charge of coarse. Be an informed customer so they don't screw you over. As I said, knowing what's wrong is a good thing but even if you're sure it's something they did, don't go in kicking and screaming. That will only cause them to go on the defensive and you won't get anywhere with them. If you do find something they did wrong, say something subtle like "I think this wire may have been routed too close to that pipe." This tells them you know what happened so they can't make up any excuses, yet gives them an opportunity to save face, do the right thing and take ownership for the mistake.
Your dome light is controlled by the door switches and the head light knob(by turning it all the way to the right I believe). Make sure the head light switch is NOT turned all the way to the right before troubleshooting the dome light circuit/problem. The courtesy lights under the dash (driver and pass side) are also on the same circuit. Do your courtsey lights work properly?
As far as charging a battery in a car, I don't disconnect the battery. If your battery charger won't charge your battery with the battery cable still connected, you've got a serious elcetrical problem such as something drawing a significant amount of current still being on ALL the time.
Is your battery old? You could just have a bad/old battery that won't hold a charge.