Can you mount four televisions on the wall and run each on a separate cable channel?

Yes, provided each TV can decipher it’s own signal. (4 digital TVs with their own individual tuners.)
Yes, using a signal splitter.
Can’t be controlled from a single remote for TVs of all one brand mounted on the same wall. The signal from remote spreads out too much.
If all TVs were different brands and remotes worked off different signals would be best but you would have 4 different remotes.
No other problems/equipment.

It depends on your cable system. Most modern cable system require a digital adapter (or set-top-box) per device. Digital tracking gives them more control over what goes where and lets them deliver video-on-demand-for-a price services.

So you can do what you want, but in addition to the splitter, you would need a digital adapter for each of the TVs.

yes provided it’s digital and you have four decoder channels.

Yes, if you are on analog CATV. Yes, if you have a digital box for each TV.