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It's possible that the filter media could clog up if there's no salt in the brine tank for an extended period but a few weeks won't hurt anything
Put about a foot of water in the tank. The tank will refill the next time the system cycles. 3/4 full of salt is OK but you don't need that much. It's actually better to put less salt becuase it's easier to break up a salt bridge if there isn't so much salt in the tank. A couple bags is usually plenty. Just check on the salt at least once a week.
Yes foot high from the bottom. The tank refills with water automatically every time it runs through a cycle.
The salt disolves in the water. it doesn't "come out". The brine is used when the softener regenerates not when you just run water. If the level doesn't drop in a couple weeks poke it with a broom handle to break up a possible ice bridge. If it still doesn't go down after that you have a problem with the softener itself and you'll need to call a pro for service.
If you have more questions about softner operation then please post a new question thread.