I have a couple of iTunes-purchased M4A songs. Purchase date is stored in PURD (tag code is also purd) and MusicBee didn't override it in the transfer. You could enable it in the custom tags section of the preferences. You'll have to search the forum to find the write-up for how to do this, I can't remember how I did mine at all.
None of the data should be lost as none of the tags are rewritten (except for metadata like last modified, date added, and such), it's just that iTunes and MusicBee handle some tags and filetypes differently.
If you import the songs into MusicBee, right-click on a song, click Edit, and click the "Tag Inspector" tab at the very top (above the normal tabs). There it'll show all the tags for that file.
Now for M4P files... M4A files are unprotected and playable in MusicBee, M4P are DMC-protected by Apple and therefore are restricted, and MusicBee can't play them. A quick Google search seems to show things that can convert M4P files, but I haven't tried any so I can't speak to their effectiveness.