NES TV screen artifacts and game behavior changing due to old console/cartridge, what is physically happening?

Whenever my nephew comes to visit I grab my NES and we play some 30 year old games. Sometimes when I first power on the console the screen comes with artifacts, removing the cartridge and cleaning the slot and contacts make the game start running fine again.

Interestingly sometimes the problem is not only limited to the screen but sometimes it directly affects gameplay, specially in the game Teenage Mutant Ninja Turtles 2 when I'm able to jump "under the screen" and can still hurt the enemies and they can't hurt me. Just to mention that this is not a pirated cartridge.

I only had the idea of making this question after some basic cleaning so I could not trigger the TMNT2 game problem, but Battle City can be seen having screen artifacts (that don't affect gameplay).

What could cause these issues in the games? Do the binary data from the cartridge to the console get corrupted and so the console use bad data to render the game? Can specific binary data be lost and change the game behavior? Is there no data consistency check done by the console over the data transferred?