Just to clarify, I mean I removed all of Lubuntu's (and LXDE) packages, as much as I could, and then install Ubuntu MATE's packages instead. The end result wasn't LXDE looking like Ubuntu MATE, but actually running MATE very close to a fresh install. 
It's possible that some old configuration was left over from 12.04 that isn't causing it to freeze... but the graphic performance was poor as far as I remember.
You could do something similar with Debian by installing the MATE Desktop packages (mate-desktop) instead of XFCE if you like. 
For your Firefox question, I suspect the enabled repositories on your system doesn't have a package for Firefox. Maybe a repository needs to be enabled?
Since Firefox is in Ubuntu, it will definitely be in Debian.