I suggest not using distilled because distilling removes elements such as oxygen, that contribute to flavor. Taste it and you'll understand. Tap water is fine. Some parents prefer to use filtered or purified water. If you drink filtered water, use filtered for your baby.
The down side of using filtered water is that we're not exposing them to tap water, which at some age they will be drinking. The more we "protect" our babies from exposure to ordinary things we are decreasing their bodies abilities to build up antibodies to protect them from exposure to the things that are in the environment that we cannot avoid. Studies have shown that the baby and child who is not allowed to crawl on the floor or play in dirt, or drink tap water, etc. are less healthy than those raised the "old" way in which exposure was not so much of a concern. My daughter's sister filtered her babies water, shampooed the rugs weekly, mopped hard surfaced floors several times a day, kept them away from anyone with a cold or who appeared ill and her babies were sick all the time. My daughter didn't do any of those things and was not concerned about exposure of anything except for chemicals and her babies were healthier.
We do have chemicals to which we do want to limit exposure but unless your city does not do a good job of clearing out pathogens there is little need to not use it for formula. There is some legitimate concern for flouride too. If you drink filtered water, then it makes sense to filter your baby's water because they will continue to drink filtered water after they're weaned. But if they will continue to drink filtered it makes sense to start with filtered.
Use whatever temperature of water that your baby prefers to drink. Yes, we are told to not microwave formula or breast milk for baby. The reason is that the microwave heats unevenly. They have to tell us to not microwave it because of liability concerns. There are some people who do not use common sense. If you shake the bottle a minute or so before feeding it and then test the temperature on the inside of your wrist there is not danger that your baby will be burned.
If you're using water out of the fridge, you can heat it first in a measuring cup, so that it's easy to pour, and then pour it into the bottle containing powder. If you're going to be away from home for awhile you can put heated water in a thermos and the formula in bottle(s) to be mixed later. That way you don't need to be concerned about the formula spoiling.