I like ur scape very nice and simple. Using tap water is not a good idea. Will give u lots of algae problem in the future. Best is to get a DI unit at the very least.
If u dun want to run skimmer also can, but u have to consider ur nutrient export. For now u dun seem to have any. I am running without skimmer also but i have a small refugium with macroalgae as well as marinepure block. Have heard good reviews about the marinepure block it supposed to help to build bacteria that can break down nitrates. Maybe can remove the biomat and replace with the block. Ur filter sock should be good enough for mechanical filter. Then the back compartment remove the bacteria media and place a skimmer