"1. Does it make a difference if I burn the music as data rather than audio?"
You wont get as many songs on the CD when burning as an audio disk, but the disk will be playable on more standalone players.
"2. Can I record these same mp3 files as an audio CD rather than an mp3 CD and if so, does it reduce the sound quality?"
Yes you can and no, it wont reduce the quality. Hopefully the original MP3s are saved at a bitrate equal to or better than 128Kbits/sec, better if they are about 192Kbits/sec although the file sizes will be a little larger.
I can't see any reason why your player should lock up though, you seem to have completed the process correctly, try playing the CD in another media player.