It's sometimes hard to believe, but our current laws seems to allow that you can be forbidden from modifying a piece of software even for your own personal use.
I think where it would come into legal question is a situation more like "modifying our software, running the modified version on a server, and making a profit from the service that server provides." In other words, modifying for commercial use.
Um, no. You have to publish or perform it publicly to be infringing. If you sign it at home, or record yourself singing it at home, or record a copy of the original at home, you're still not infringing as long as you don't attempt to distribute it.
Unless your shower is located on a stage and you charge admission for people to listen to you sing in the shower, but I'm pretty sure that's not what you meant.