On the regulator, I've always put the dust cap on the first stage tightly, then submerged and and/or rinsed the entire regulator in fresh water. In 20+ years of diving, I've never had any problems with water entering the unpressurized system. It's important not to press the purge button on the second stage, and make sure the cap is tight on the first stage. Doing this, I haven't had a problem.
If water enters the system, you can sometimes see the pressure gauge fogging up, and of course corrosion in both first and second stages will increase and could lead to failure. I have seen - many times - divers forget to put the dust cap back on the first stage after a dive, return rental regulators without the caps on, or hose them off without the dust cap not in place. Not good. Make that the first thing you do when removing a regulator from the tank - dry off the cap and replace it on the regulator.
Salt is completely soluble in fresh water. A nice soaking and rinse will get rid of the salt very effectively - no chemicals needed.
When diving on a charter boat and doing several dives a day, I try to rinse off my gear soon after the dive, but if I don't get to it immediately, no worries. Soaking my regulator or other gear immediately after a dive in not necessary, in my opinion. I'm probably using it within a few hours anyway, and any salt on it will be gone (dissolved), replaced by new salt!
When shore diving, rinsing with fresh water immediately after a dive is not always practical. On the regulator, be sure to replace the dust cap, pack up the rest of your gear, and rinse it when you get home. It's not the end of the world.
As for other gear....
Mask/Fins/Snorkel - a thorough rinse or soak in fresh water
Tank - hose it off thoroughly, paying attention to the valve and boot area
B.C. - Soak or hose it down on the outside, and add some fresh water to the inside through the oral inflator. Swish it around, and drain it all out by holding the BC upside down and draining it out of the inflator hose. Do this several times, and store the BC partially inflated. There are some cleaning solutions for the inside of the BC that might be worth while, but I've always used fresh water.
Hood/Boots/Gloves/Wetsuit - It's best to soak these in a tub of water, and wetsuit shampoo makes them smell pretty and is probably better for the fabric. Make sure they're completely dry, inside and out, before putting into storage.