Lots of advice and potential problems here. First: triage order of priorities:
Ammonia: I noticed you do not have a very sizable amount of rock in the tank. You might want to add some additional dry rock (be careful about adding live rock at this point) to give greater surface area.
RO/DI water: You will save yourself many headaches if you can find yourself a basic RO/DI (4 stage) filter and put it in parallel to your laundry tap (or just periodically remove the hose from the laundry machine and attach the filter to that outlet). Remember you have to have an outlet for the "dirty" runoff water, either into the washing machine itself or into your effluent drain. (The "dirty" water isn't really dirty for use in washing clothes, it is just concentrated tap water).
Nitrite (NO2): At those levels, do not worry at all! In a freshwater tank I would say otherwise, but those low levels of nitrite will not be a problem for marine creatures. Eventually you will see the nitrite come down as the bacterial colonies get a foot hold, but there is no reason to try to manage that level. (Old school guidance would have told you otherwise, but studies a good decade or so back show that there are minimal to no effects at low concentrations, only when they get on the order of 100ppm or so). Read here to verify:
http://reefkeeping.com/issues/2005-06/rhf/index.php
Right now I would be less worried about nitrate and phosphate then getting your biological filtration stabilized. If you never added bottled bacteria it may not be a bad idea to add a bottle of Fritzyme TurboStart 900. There are others (Tims, etc), but that is the most recent one I had success with. But regards nutrient control in the short term since you have a small tank (15 US gallons, I myself have a 14 gallon cube), you could do well once you have a good water source (DI filter) to do weekly very sizable water changes for a bit. I do 80% changes weekly, enough such that there is only an inch or two in the tank to allow the clownfish to stay upright! Now only do this if you have the confidence to match the salinity and temperature very closely to that of the tank's. I use a metric scale to measure my salt to the exact gram (then verify with the refractometer once mixed) and I use an identical heater to what is in the tank set to the same temperature. It will never be EXACTLY the same, but it should be pretty close. You may want to start with smaller changes until you are confident with your process, say 25% to start, then 50, then whatever you are comfortable with and which don't show stress on the inhabitants. One of the local fish dealers here has a 7.5 gallon nano for which he takes the level all the way down until the hose looses suction, and the tank looks great! (Mind you he is using repurposed water from the shop's other coral tank systems to refill). Also, if you start doing such large changes, do yourself a favor and never rush the mixing.....allow the new water to sit overnight each and every time.
One thing that large level a water change allows you to do is to rely less on testing and manual tweaking (two part dosing, etc) to maintain suitable Alk/Ca/Mg levels. Once again, these (Alk/Ca/Mg) should be secondary to the ammonia, but given the choice of doing a one hour water change per week versus spending that time testing, calculating dosing levels, etc, I would choose the changes. If your tank was 600 liters, that would be a different story of course. A few other tricks: feed less! Let me say it again: feed MUCH less (for now)! Your fish will survive on two to three very small feedings a week, and for now that would be a good idea until the tank matures. On the day you are going to do a water change, that is the day to feed your corals and give your fish a slightly (slightly) larger feeding, then do the water change within a few hours afterwards to capture all the leftovers. Also, before you siphon out the tank's water, really stir things up either with a turkey baster or simply using a powerhead and blasting the rock (and stirring up the sand to release detritus into the water column). You can cover the inlet of the powerhead slightly with your palm to control the outflow blast/rate.
One other thing, you should elevate the skimmer up a bit higher in the tank to skim a bit "drier", as right now it looks excessively wet. It takes tiny adjustments and then waiting a day to see how it responds (you can never judge right away). No problem if you take it too high and have a thick sludge at the end of the day, but a coffee or lighter tea colored skimming is what you should aim for.
Hope this helps a bit.
But after a couple months of working with it you will get a feel for how thick you can pull the skimmate and you won't imagine having a tank without one after seeing the muck it can pull out. Also, since the tank is still relatively young, there may not be a lot of organics in it yet, though as the doctor mentioned, you should start seeing some.
