Can I trust that my Geiger counter (GQ GMC 500) is in fact measuring radiation?

I had a look on a Radon map, and you are in an area with very low Radon. That's good. I find it very interesting that you 'feel' this phenomenon. Radiation has no sensation whatsoever. One of the reasons it is dangerous when close to very high sources, without proper protection or detection. The levels you are seeing from your counter are not even worth concerning yourself about. The hot spots are probably NORM in your house construction, or other items/ornaments. Essentially everything is radioactive, to some degree. 

Using the Radiacode with those levels that you are seeing will be pointless. You won't get any good data. Even with weeks to months of samples, the noise would ride over any signal. As NORM has many different isotopes, these will just create counts all over the spectrum and essentially give you a 'background' spectrum. If you purchase a Radiacode purely for +20 cpm background at your home, you will bbe wasting your money. You are going to need 200+ cpm (with your counter) and a couple of weeks to identify anything for certain. 1000cpm and maybe a week. The Radiacode is orders of magnitude less sensitive than what you would require to work out the exact isotopes that are giving you +20 cpm.

Going back to you being able to 'feel' this phenomenon in certain areas of your home, have your considered RF/EMI from electrics/electronics around your home? If you have a lot of electronics going on, and I am including basically everything that runs from mains, you may be better putting your money into an EMF meter such as the GQ EMF-390. I have a couple of LED light bulbs at home and they my EMF meter go absolutely nuts. Some people are sensitive to EMF, some people aren't.