I would like to use a set of 105 shifters (10 speed), but I don't have a rear mech yet for them. I would like to purchase a mech that would allow me to use a big cog cassette, like a cyclocross or mtb one. What is the compatibility with other models, like mtb mechs? Thanks in advance
It depends on what you mean by big, which can now be interpreted many ways.
Without doing anything too tricky, you can run them with any 9-speed Shimano mountain derailer, some of which can natively handle up to a 36t, RD-M592 being a currently made and easily available example.
10 and 11 speed Shimano mountain rear derailers use different actuation ratios, so they can't go directly on your shifter, but they could be made to work using something like a Jtek adapter or Wolf Tooth Tanpan. If you wanted to run a 40 or 42, you could use a Wolf Tooth Goatlink on an applicable 10-speed RD that you also have a cable pull adapter on.
If you want to use a road RD, you can go up to a 40 with a Wolf Tooth Roadlink. It's important to understand that they increase the clearance but not the capacity of your RD, so whether it makes sense depends on what's going on up front. (True no matter what route you choose actually.)