1. Should I use a heater?
That depends on the temperature of your house. Buy an aquarium thermometer and if the water registers below 75 degress Fahrneheit then you need one. I would recommend a Stealth Heater by Marineland, they're really reliable. If you get a different kind, just make sure it's adjustable and shatter-proof.
2. Should I use sand, or just rocks in the aquarium?
I assume you mean for the bottom? Sand is better than rocks because sharper gravels can scratch up their bellies. There's also the risk of them swallowing a small rock during feeding time. I personally have a bare-bottom tank (nothing on the bottom) for ease of cleaning
3. What is all this about gravity and hard water? (New to this stuff )
You're probably seeing specific gravity being mentioned because mudskippers are brackish. It's a measure of the density of the water compared to regular/distilled water. You can measure it with a device called a hydrometer (sold in pet stores in the marine aquarium section). The reason this is useful is because you can estimate the salinity (salt content) of the water since saltwater is more dense than freshwater. Mudskippers are brackish, which means they live in areas where freshwater and saltwater mix. It's kind of an in-between area, if that makes sense. While a marine aquarium will normally be kept at a specific gravity of 1.021-1.025, a brackish aquarium should be lower. A specific gravity between 1.005 and 1.010 is usually maintained, but you can raise it higher if needed. What all of this means is that to keep mudskippers, you need to make brackish water. You achieve this by mixing a marine salt mix (I'd recommend Instant Ocean) with dechlorinated freshwater. Once you've added some salt and mixed it in, measure the specific gravity. If it's within the correct range, you can add it to the tank. If it's too high, pour out some of the water and add pure (but dechlorinated) freshwater. If it's too low, add more salt mix. The great thing about brackish tanks is that you can afford to be off a bit. Brackish animals are extremely adaptable and can handle different specific gravities. Some people even feel it's beneficial to have it changed every so often, as opposed to the average marine tank which needs to be kept at the same S.G. every time you do a water change. Honestly, I often don't even measure the S.G. when mixing water for one of my brackish tanks
. But that's partially because I know from keeping marine fish how to estimate salinity.
Water "hardness" is a measure of how many divalent ions (like calcium) are present in the water. For fishkeeping, all you really need to know is how to measure it and maintain it at appropriate levels. Buy a chemical test at a pet store to test it, rather than a strip test (strip tests are less acurrate). You'll probably just want to get a basic test kit that also includes tests for ammonia, nitrites, nitrates, and pH. If you've kept other fish in the past, you might have some of those tests already. The test booklet should tell you the correct levels but don't worry about it too much. Your marine salt mix should keep both the water hardness and the pH at acceptable levels unless you slack off on water changes or have extremely soft tap water.
4. What should I feed them?
At first, just feed live foods like crickets and small earthworms. You can then try offering frozen foods (using tweezers to move them around helps "trick" the mudskipper into eating them) such as bloodworms and krill. If your mudskipper will eat pellets, you can offer those, too. They won't normally touch flakes and those wouldn't be a balanced diet anyway.
I hope this helps!