Mystery valve on hot water line from water heater
It's not exactly like mine, but it looks like it's probably an overpressure (no temperature) type valve (aka pressure relief valve), and it should not be dripping unless you have overpressure - so a constant drip means it's likely in need of replacement. (Or possibly just cleaning, if you get lucky; don't expect to get lucky.)



Indeed, it looks very much like this adjustable one.

The line on the cold water line could be intended to feed a hose bibb.

Given that it's probably intended to relieve pressure events before the P/T on the tank so that the P/T on the tank does not become leaky through frequent activation, you probably could replace it with a pressure tank.