7805's are linear regulators, they drop excess voltage as heat. Using a 2A rated 7805 won't make it run less hot, the heat output is the delta between input and output voltages.
Not necessarily, the higher rated packages are rated higher because they use higher quality materials that have a lower thermal coefficient (for getting the heat out), so yes the energy in Watts is the same but they’ll still probably run cooler.
If you’re interested whether this is the case you’d look for a thing called “junction thermal coefficient” on the data sheet.
It's been ~15 years since I did much with 78xx's, but I seem to recall the various package types were all pretty much the same on thermal coefficients, and other related specs. It's a decades-old design that has been pretty well optimized at this point. The 2A versions also have slightly bigger leads, which may not fit in through-holes not sized for them originally.
Are there not 7805 replacements which internally are switch mode? (It seems almost miraculous to me that we could miniaturise a switch mode power supply into such a tiny device.)
