To answer your questions:
1. Moving to one page to another
In general yes, going from one page to the next makes the music start over, or at best (if you use javascript to keep track of where the user is in a song) the song will stop for a few seconds while the new page loads and then you can skip to where they left off.
There are a couple of work arounds but none of them are pretty.
You put your site in an iframe or frameset so whenever the user navigates in the frame only that part of the site loads so you can have part of your site not refresh
Convert all your links to ajax so whenever a user clicks on a link you use ajax to query the page and replace the main content div of your site with the data you retrieved from ajax
Have a pop up. Most browsers block pop ups but if the user lets you you can have a separate window with your music
2. Performance issues
There aren't really any performance issues with music playing. Browsers pretty much have it down great. As far as I can think off the top of my head I can't think of any way to really improve performance. Maybe reduce the quality of the .mp3 so it's smaller or change it to a newer format that's smaller?
3. Music list
Yes it's definitely possible to have a list of files that the user can choose from. Streaming music from a different site is usually not allowed by the other site. You will have to check with them to see if they'll let you and if they will they may have an api or something that would help you with a way to list the music. This question is a little too broad to answer on SO. If you want help with creating a music selector try making one on your own (most likely a select box that has an onChange listener that then changes the source of the audio player) and if you can't make one on your own then post the code you've tried and people will be glad to help
In general
My personal opinion is I hate sound/music or ANY noise coming from any web pages. Sometimes it bugs me so much I close the page and never go back. The reason is, I have my favorite music or TV show that I'm already listening to and I don't want to be bothered by some site. That is just my personal opinion but I imagine that it's a pretty widespread opinion since none of the popular sites (excluding music-centric sites like pandora, grooveshark, etc) have music playing in the background.
I hope all this helps you.