Can I somehow make the LM7805 work hand in hand with an N-channel MOSFET to get more current out of the regulator?
Whether you use a PNP or a MOSFET, you will be producing the same (more or less) amount of heat.
Linear regulators are nothing more than automatically adjusted variable resistors.
When you use a PNP transistor to provide more current, the transistor is just playing the part of a higher wattage resistor.
If you drop 12V to 5V at 5 amperes, that's 35 watts of power wasted as heat.
It doesn't matter if you use a PNP transistor, a MOSFET, or Maxwell's demon cranking the wheel on a tiny rheostat - you have to dissipate those 35 watts.  Something is going to get hot.
If you want to produce less heat then you should look into a buck converter.  That's a switching regulator that uses inductors and/or capacitors to lower the voltage with very low losses.

Swagatam shows how a MOSFET can be used with an LM317 for higher current output.  The same concept will work with the 7805.
It probably doesn't work well.  Don't come crying for help when that circuit explodes or eats your hamster.  It's a bad idea, and I make no promises about how well designed it is.