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

Just to add to what ChrisLX200 said, there are very few places in the world where you would be able to a radioactive dose that would actually harm you in a short time. There are even less places you could go to like that, that are open to the public. The only places you could get a dose enough to harm you in a short time are nuclear material processing plants, specific labs and nuclear power plants. It always comes down to dose x time. The higher the dose and the longer the time, the greater outcome of harm. Once you get out of the instant or short time harm time frame you are getting into long time exposure, which generally will result in cancer, some time on from initial exposure. To become ill and start showing symptoms will take a tremendous dose of radiation in a short time, other than that it would, in many cases (unknowingly), cause cellular damage over the length of time exposed, and thus lead to DNA damage and cancer.

With only peaks at ~40cpm, getting a Radiacode would be a waste of money if all you were intending on using it to find out what the extra 25~30 cpm are. The counts are too low and the Radiacode will just pick up noise (NORM/cosmic rays etc). You would need to be in the hundreds of cpm and then leave the radiacode for several days to identify anything. The signal to noise level will simply be too low. If you are getting spots in your house with increased counts it is most likely radon and it's decay. A radon detector ( such as the Airthings Corentium ) may be a better choice. Obviously this will only let you know if Radon is present but will give you a clearer answer than the Radiacode with the low counts. You could also try doing a wipe test in various places of your house. Besides man made sources of radiation, Radon accounts for a good portion of the radiation on this planet, the other good chunk is NORM. The chances are it will be one of those two, or a mix thereof that you are finding. 

You are comparing counts inside your house to average levels. Where are you getting the average levels data? If this is public/government data then the readings will be taken outside. Radiation levels are generally always higher inside a building than outside. Also do you have any historic counter readings from these hot spots, or is that you have not long since got your GMC-500 and have recently noticed the ~40cpm? It would be handy to know if you have any historic readings, and also what is the cpm in the areas that are not hot?