I hope your Betta is doing great in his new tank. Just want to add a few options on how to acclimate new fish.
There are a few different ways to acclimate. You can float the bag with the fish in the tank so the temperature can get equalize. You would then take a cup of tank water out of the bag and replace it with a cup of tank water. You repeat that every ten to 15 minutes 3-4 times. This will give the fish enough time to get used to the water parameters in your tank. You then can either let the fish swim out of the bag into the tank, or net the fish if you do not want any of the water in the bag get into your tank. The reason why this is advisable is, that you would reduce the chance of bringing anything into your tank via the LFS tank water. Others may argue, that water from the LFS and diseases may still be on the fish anyways so it does not really matter and you could empty the bag into the tank after acclimating the fish.
In your case it is just a Betta going into his own tank, so there are no other fish that are at risk. If you were to introduce a new fish into your community tank, it would be advisable to quarantine the new fish for about 5-6 weeks. If the fish has been infected with something it would show up during that time. It is a lot easier and cheaper to treat one (or a few) fish in a quarantine tank than a whole community where possible fish deaths could get very costly.
Another way to acclimate is doing a drip-acclimation. You would carefully empty the bag with the fish into a small bucket and run an airline tubing from the tank to the bucket. You fashion a knot in the tubing, by tightening or loosening the knot you can adjust the speed of the drip. You start a suction in the tubing so the water can start dripping from the tank into the bucket. You can cover the bucket to darken it, that will keep the fish calmer. In 60 - 90 minutes you net the fish out and place him into the tank. The drip-acclimation is the better option for very sensitive fish, or if the tank water from your LFS differs greatly from yours. That may be the case if you are getting the fish from a different city, that is on a different water supply.
You brought your Betta home in a cup. In this case I would have emptied the cup into a container that is a little larger and added about a third more of water. Every 10 minutes I would have added some more water, and then net the Betta out to place into the tank. The mossball will be fine with a few occasional squeezes in tank water to clean it out. Food and waste particles like to settle on and in it, so cleaning it during water changes is a good habit to get into. I hope all goes well with your tank. With a few water changes any salt residue from the substrate will lessen like Tall Tree already mentioned.