1) for water and salinity testing I recommend you get a good refractometer to test and not a hydrometer, most of which are not very accurate.
I totally agree about your suggestion to get a good refractometer and also a bottle of 35 ppt calibration solution.
2) for your tap water, what you need to test for is Total Dissolved Solids or TDS. Unless your tap water is coming out at zero TDS, it's unlikely you will be able to use your tap water for long term success. It's good that you're testing 0 nitrates, but that's not enough. You really need to be using RO/DI water to mix your saltwater. You can either buy it at an LFS or most of us get a small unit that we install in our homes and make water at home which is cheaper in the long run.
Just trying to help.
My LFS does not charge for RO/DI water and even if his store does the OP should tell them they should absorb the charge as a customer accommodation after all he was put through. He mentioned that he lives only 2 blocks away from the LFS and considering how small his tank is it probably wouldn't make sense for him to invest money and install a system in his home right now.
This water issue, however, is not the most important issue right this second. Your more important issues are getting the ammonia down in your tank (change some water) and treating your fish if they are sick, as opposed to just having gill burn.
I totally agree, this is extremely important.