Can having a MBP constantly charging damage the battery?
Constant charging should not damage the battery in an extreme manner. Yes, you would get more life from it if you would let it always cycle between 40 and 80%, but the Mac is already cycling power in the high 90s even though it is not showing it to you. I have not been cautious at all with by 2009 Macbook Pro and I can still get around 2 hours of autonomy from it.

However, a bulging battery is a very different scenario. You should get that replaced as soon as possible. Several people (on forums) have reported that they got their batteries replaced for free, even in out of warranty computers. I would strongly suggest bringing your computer to an Apple store if you can.