So I just got an APC20 and I am having a huge number of difficulties with Ableton Live. I was wondering if it is possible to connect this, or any other sort of thing like this (Novation Launchpad) to a different software, for example LMMS.
I can't speak for LMMS but it absolutely is possible to connect any midi device to any program that understands MIDI. Getting them to do what you want can be very challenging depending on the MIDI controller and what it was intended for.
The APC20 was made for abletons session view, launching clips is its primary purpose. I know LMMS does not have such a feature so don't expect it it to work the same way. You should be able to map buttons to simple commands on the LMMS control surface, stuff like transport controls and mapping volumes(or other parameters) to the volume sliders on the APC20.