Any milky stuff that collects on the bottom will be precipitated Calcium carbonate and any heavy metals that may have been contaminating the lime. This happens when the limewater comes in contact with Carbon dioxide and cannot be easily prevented.
The issue with adding Calcium carbonate particles is that they act like seed crystals when added to a supersaturated Calcium solution (your saltwater) and will actually pull Calcium, Alkalinity, and Magnesium out of the water as the crystal grows. The Magnesium will quickly stop the crystal from growing but it does not prevent it from starting. A few particles is't going to have much effect, but if you do it a lot then it will all add up.
Limewater raises PH by removing excess Carbon dioxide from the water so if you have other issues it's not going to help.