Almost. Try opening the laptop up. When you reach the battery on the motherboard, remove it for about 15 mins. Reinsert it, assemble the thing up and try rebooting. If that fails, then I would say yes, you're screwed (unless someone comes up with a better solution).
Another thing would be to send the laptop for servicing if it is still in warranty.
Nope, you'll need to solder it back on. otherwise, it could fall out during operation and short out your mobo. There are conductive glues out there, like cold solder, you can try those.
After you take the battery out, though, don't solder it immediately back on. make sure the thing boots up before you do so, you don't want to have to do it over again.
The time interval to keep it out varies from a few mins to a whole day, according to various forums. just pick a median time and go with it.