I was the manager of a wet lab when I was in Hawaii, and it was stocked with equipment from Hach. Most of their tests are EPA-approved for use with seawater. I invested in a Hach colorimeter, digital titrator, heater block, and HQ40d multi-meter to test for:
Alkalinity
Ammonia
Calcium
Iron, Total
Iron, Ferrous
Iron, Ferric (by calculation)
Magnesium (by calculation)
Nitrate (approved if corrections are applied)
Nitrite
Phosphate, ortho-
Phosphate, total (requires digestion)
Phosphate, acid-hydrolyzable (requires digestion)
Dissolved oxygen (HQ40d)
pH (HQ40d)
Temperature
I also have the Hanna Checkers for alkalinity, calcium, phosphorus, and iron. These produce results that compare favorably to the expensive Hach equipment.
At one time, Instant Ocean marketed comparators that used Hach reagents. Not sure if they still do.
I've used Salifert's alkalinity test - its results were favorable to the titration method.
I use the Red Sea potassium test. A lengthy test but it reports numbers that are at least ballpark hence I tend to trust it.
For specific gravity, a refractometer is the way to go, IMO.