I'm not sure if one of your questions was answered, so I'll answer it.
You asked if putting aquarium salt in is okay.
Aquarium salt is good short term for healing external issues (1 teaspoon per gallon). After ten days of use, you have to discontinue it or else your fish could suffer from damage to the organs. Honestly, you're fine and clean water without aquarium salt will do well.
Now that he should be getting better water changes, there shouldn't be any reason why he would tailbite (unless there is stress). Hopefully it'll be a one time thing.
As for testing your water, if you want to invest in a test kit, go ahead and buy a good quality one like this test kit
. It's a bit expensive, but it lasts forever (mine lasted for about 2 years), and it gives very good and reliable parameter readings. :) After a while of testing (maybe every day for a few weeks), you'll be able to see how fast and high the ammonia in the water climbs before your weekly 100% water change. If the ammonia consistantly gets higher than .25 ppms before your weekly water change then you know you need to start doing partial water changes as well midway through the week. However, in a 3 gallon, a 100% water change once a week ought to be enough