I have the Phillips Expanium MP3 player that reads CDR's and Audio CD's and most mornings I use MJ, exactly as you step by step describe, to make "Today's Music". It burns while I have breakfast.
It plays on the Phillips while I pushbike to work then into the CD drive and play in MJ for the rest of the day.
My hardware is nothing special, I have WinME on the PC that burns and its a bit unstable, but I don't remember that last coaster that MJ burnt.
In short I have no idea why you're having the problems that you're having.
Can you get someone else to burn some MP3s onto a CD and see if that works in your player?
Or use some other software yourself to burn an MP3 CD to see if that helps isolate the problem.
Have you tried checking the "Use Generic Driver" box under Options?
Post some info about your system, your burner, your OS and version of MJ.
In MJ, Help, System Info - copy and paste here. There are some clever folk around who can help I'm sure.