No. A trademark is to tell a customer that a certain good was made by a certain person or company. This is prevent confusion over which company makes a thing. It's why you can't create an operating system kernel and call it Linux. You can create a clothing company and call it Linux though because no one would be confused that Linux clothing is in any way related to the computer operating system component.
Band names can and are trademarked. So you can't call your new band "The Beatles" or "Five Finger Death Punch". Titles are subject to copyright though, which is a separate thing altogether.
Definitely not. There's a whole genre of music that's created for riding the coattails of popular songs. They wait for a song title by artists like Taylor Swift to be announced and then release their own songs with the same title. Sometimes they're actually good, like this dude:
The general consensus of the internet seems to be no, although this surely varies to some degree based on the laws in whichever country you're in.
Before anyone tries the other avenue of attack, titles to things generally cannot be copyrighted, either. Content of a work can be, but the name of it cannot.