This is the solution I'm using, no conversion necessary:
The resulting embedded flash player is customizable by changing the parameters. You can change settings such as autoplay on/off, player colors etc. You can learn how to do that here MP3 Player
Code generator page is here MP3 Player - Players - MAXI - Generator
6. The flash player will appear in the post.
That appears to use an external site. Which would work I guess, but I prefer not to rely on other sites to provide my content. Best case, it slows load times. Worse case, the other site is down. Worst case, they decide to start cookie stuffing, flash style.
The benefit of it of course is it allows your users to upload files and you can possibly integrate and play them dynamically. If that's an aspect of your site then I guess it would work, although personally I'd prefer to install the software it on my own server and run it from there. If I have pre-determined content, I just convert and stream it normally.
As for server side elements, you dont need anything on the server for swf. You just upload a swf file, and point to it in html and it loads just as easily as an image. You literally just load the object, and the flash player at the client side will handle it.
If by player you mean the little thing with the buttons (play pause etc) that's just another SWF file that is told to interact with the main swf. I believe the flash player has this functionality anyway if your embedded object has a width / height specified but no viewable content to put in that real estate. I'm not sure though, I've always used my own "player".
As for streaming; Flash content is streamed.