Why is it damp?
Because it has not been refined or kiln-dried, and there is no anti-caking agent in it. A little moisture is the sign you are buying the unrefined thing rather than a fault.
Is it healthier than table salt?
It contains trace minerals that refined salt does not, but the sodium content is broadly comparable. It is worth buying for the flavour and the way it is made rather than expecting it to behave differently.
Can I use it in a grinder?
Yes, with the coarse grade and preferably a ceramic mechanism. Damp salt will corrode a steel grinder over time.
How should I store it?
In a sealed jar away from steam. It will clump if left open in a humid kitchen, though breaking it up with a fork does no harm.