I couldn't get it to play either without downloading a Flash Player 11. Perhaps it wouldn't play for you for the same reason. I did watch the video by downloading it direct, just the .swf, not the page.
In my player there was no sound and the video was pretty grainy. Maybe that's just because my player is not a version 11, but it could be other things.
First, to maintain the highest quality, be sure to create the .mpeg at higher bitrate than what you expect to encode it with the Flash Video encoder. So for example, do the .mpeg at 1000kbps and the .flv at 500kbps. You won't lose as much quality with the transcoding that way.
Second, just use a Flash Video player embedded into the page to play the .flv file. Skip importing it into Flash to make a .swf. The .swf will play in a browser with the the Flash Plug-in. But use a Flash player and play the .flv directly instead.
If all you are doing is using a .swf, then the .flv and the .fla are not needed by the Web page, don't upload.