parameters I need to test for - I usually go by the look of my other corals. The tree coral is a good indicator. (I make weekly water changes anyways)
If you plan on keeping stonie corals. thats going to have to change.
Those snails are probably eating the algea that is building up on the bare skeloton. they are not the problem.
I wish I can help more , but unfortunatly your don't know your water perameters.
first thing I would suggest is frag it. break off a good portion of the coral that still has tissue and polyps and super glue it to the rock. then go buy alkalinity, nitrate, calcium, and PH test kit.
How old is the tank?
do you have any other SPS corals?