Hello Proteus, and if I haven't already zapped you then:
The issue of protein skimming a nano has pros and cons. A pro is that it can result in fewer water change requirements, oxygenate the water very effectively and also reduce algae infestation risks by pulling out nitrogenous compounds from the water. Some of the cons are the price, the extra plumbing and pumping of the setup, the storage space etc.
Basically Id say if you want to skim one that is definately fine, but for the bioload you have described I think your tank will be okay without one.
What you mention about live rock is generally this: Live rock acts as a 'filter' {in a general sense} by providing surface area (tons of it depending on the porosity of the rock) for nitrifying bacteria. These are the organisms that convert harmful ammonia in a series of events to less harmful nitrate, a fully-oxidized form of nitrogen that does not 'burn' or react to pH of the water as primary ammonia/ammonium does.
Live rock simply oxidizes the nitrogen compounds we have in solution to a less harmful form, nitrate or NO3. The actual work is done by aerobic bacteria, but they need a large surface area to live on and this is where LR comes into the picture. NO3 is still very usable fuel for algae though, so we try to deal with the buildup of NO3 through water changes most often, then macro algae refugiums, then reduction equipment or sandbeds (much less common in nanos than in larger systems).
What are you nitrate measures each week? A skimmer can help reduce this. I think Matt likes the CPR bakpak series and I read a lot of good input on this for a ten gallon tank. Have you tried 50% water changes bi-weekly to keep your nitrates in an acceptable range (unskimmed Id shoot for 10-30 ppm)? If this is too much hassle then a skimmer may reduce maintenance hassles.
Does your system ever register a reading for nitrite or ammonia? With the one fish you've described Id guess it does not, so in my opinion I think you should buy a skimmer if you want to be able to go an extra week or two in between water changes. Sorry my response was so lengthy, but thats whats floating around in my head when I think of skimmer/no skimmer for a nano.
Welcome@