No....apps that need root access are called root apps. These include Clockwork Mod and Titanium Backup. Some tethering and screenshot apps need root access. I believe all ROMs need root access.
All apps that could be installed prior to root can still be installed after. I don't know of any specific music app that requires root. I personally like WinAmp and TuneWiki Pro music players (neither require root) because I like the scrolling music lyrics.....
I cleaned this thread. If you can't be polite & help, then move on. Not all the posts I deleted where rude. But since I deleted some that where, the others where deleted since they wouldn't have made sense by themselves.
Everyone will give a different definition of the best root apps. But here's some I use. Some are paid apps, some are free.