1. Fast boot: yes, but you should do some tweaking. If you just do a standard redhat or mandrake install you end up with a lot of services running you actually don't want.
2. First install linux and then a distro: No, you will always have to install a distro. To get an GUI you must also install XFree (that's what most distros will automatically do)
3. Paying: No, plenty of distros are free to download. Lindows is not free, it sucks. Normally you will pay only for the CD's, manuals and support if you buy it.
4. No Winamp: As said before, XMMS is a perfect alternative.
In my case I made my own "distro" it is only 7 MB and based on slackware 8.2. But it is without an GUI (I have an HD44870 charater display connected to the parallel port) and highly customized, which makes it pretty useless for other people......unless you know linux pretty well.
Here is a site with that reviews distros:
http://www.distrowatch.com/
On my home computer I use Slackware 9.0. And Windows XP for the games only