Older CD players don't 'have a problem' with MP3s - they simply aren't designed to play them! MP3 is a file format that wasn't around when CD players were first developed.

A CD player that is designed to recognise and play MP3s will usually have the MP3 logo on the front, though not always.

A very handy CD player is the Teac CD-P650, which will play audio and MP3 CDs, but also has an USB slot which will recognise both USB fobs and iPods - and will even control the iPod via its remote control. I've been selling these to hotels, churches and the like for a while recently, and nobody has had a bad word to say about them!
(cue deluge...!)

As others have said, burning an audio CD using MP3 files as the source will not result in an further losses, but you cannot rip an audio CD to MP3 and expect to have no losses if you then burn an audio CD from the MP3s - unless you have specified lossless MP3s in the ripping process - such a format does exist, though the files are roughly the same size as WAV files, so somewhat pointless!

Typically, an MP3 of a WAV file will be around a tenth of the size, so a general rule-of-thumb is that you can get 10 MP3 albums on one CD, give or take - BUT you must play the resulting CD in an MP3 compatible player!

My brane hurtz...