Can I use logic with a touch screen monitor?
no. unless you feel like writing the software yourself.
What are some ways I can use logic without touching the mouse too much?
you could try a trackball instead of a mouse. or you could try using a logic control surface, or some other kind of control surface (there's a bluetooth one available that works with logic, can't remember the name sorry) which will let you control the transport and lots of other mixer functions with buttons and faders.
also, I use a griffin powermate. it's a bit gadgety and doesn't do all that much, but I use it as a jog wheel for the logic transport, and it works pretty well for that.
and also.. have you not noticed that logic has a list of like 65 billion assignable keycommands? ok, not that many but almost.. spend some time learning the ones already assigned and then assign your own. I use quite a few and keep adding more as I think of new things.