or more importantly, how to protect your work from other "artists" claiming it as their own.
I feel like this is the kind of thing you just stumble upon, or else your fans do it for you.
To protect your music from people trying to take your rights from it or claim its their own: the minute you create something, you own the copyright to that, and unless you sign a contract signing AWAY that ownership, it will be yours until the end of your life +70 years. However, that is difficult to prove. As Steven Gold said below, registering with ASCAP, BMI, or the copyright office gives you absolute proof of your ownership for a fee.
To protect your music from labels and sketchy contracts: First, get legal counseling for a contract. It's important. Secondly, consider whether this label is worth it. You are signing them off the rights to your Intellectual Property (IP), which means you can no longer use it freely (in commercials, for example). If you want to exploit this piece in the future, then reconsider. Thirdly, consider if you can negotiate for the contract to be more accommodating for you as an artist. I'd recommend sending back a contract clearly illustrating that you are giving them PERMISSION to copy, publish, and exploit--but be very clearly that it is permission, and you are not reassigning your IP rights to them. This way, you can still have all of your IP rights to your songs and use them flexibly. Remember: labels exist because of you. If labels die, musicians still exists. If musicians die, labels have nothing to sell but remastered editions of old shit (hint: they die too).
To protect your music from free downloaders and pirates: essentially impossible. If it signed to a label, let the label handle it. Honestly, one of the coolest things I ever saw was one of my songs on the Pirate Bay with ACTUAL seeders. The fact that people would go through the effort to torrent my music was actually flattering.
Once again, pay for an hour of a lawyer's time to look over the contract. Don't just trust this anonymous internet advice.