Honestly the quality of a mic comes from the room it's in. You can have the most expensive mic in the world sound creppy just because it's in a bad room. I can get the mic on my $20 headphones to sound professional just because I set up the room I record in.
I don't exactly know what you mean by the second question though. I assume you mean listening to music while you livestream with a mic? If so, it's not a problem. Input from your mic and your desktop are separate and should abide by their own individual settings on whatever recording program you have. But if you want to listen to your own background music, you'd have to do it with headphones or else your mic will pick it up.
For cancelling background noise, changing your settings and environment would be more effective and cost-efficient than getting an expensive mic. There should be settings on your computer, but if you have a streaming program like OBS, there will be settings there as well. I also use a separate (free) program where I can make more accurate adjustments to my voice, which you can find in the tutorial here: https://www.youtube.com/watch?v=2VKxbYt1-mw
You can also find more info about how to set up a mic-friendly room here:
