Once amonia gets going, does is get broken down into Nitrites and then into Nitrates?
Yes, when am and nirtrites are zero, teh cycle is over. Do a water change.
Are the Nitrates harmful to the fish?
Yes, so are am. That's why you shouldn't cycle with fish! It's cruel and mean. Though damsels are hardy, you may lose a few. Once your tank is cycled, try to keep nitrates under 20.
Can high Alkalinity be harmful to the fish?
Yes. High or low alkalinity and ph is harmful.
If your sg is getting higher, then either your addind saltwater as top off water, or you're not adding top off water. Water evaborts from the tank, use fresh water to replace this water. When the water leaves the tank, the salt stays,
giving you a higher salinity level.
Welcome to the site, sit back and read. There are MANY threads on this. Check this forum
https://www.saltwaterfish.com/vb/forumdisplay.php?f=14 . Archives & FAQ. And this thread,
Forum Advice for New Hobbyists
Ask before buying, and don't listen to your lfs.