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

Thanks, Simomax, This is quite the explanation. Yes, I did purchase the Radiacode. After a week of using it in the house, I picked up the following list of isotopes, ranging from high to low counts (Enough data has accumulated to distinguish between peaks and background). Note that I did not include all the emission lines of these isotopes.
Iodine-131 82k
Lu-176 84k
LU 177 22k
Sm-153 78 k
Am-241 67k
Eu-152. 10 k
Tc-99. 45 k
Pb-210 Chain Ra-226. 37 k
U-235 Chain U(235-238) 28 k
Ba-133 40 k
Pb-212 15 k
Pb-214 Chain Ra-226 7.5 k
Ac-228 Chain Th-232 4.25 k
Annhiliation Line 1.5 k
Kr-85. 1.25 k
Ti-208 Chain Th-232 1k
Bi-214 Chain Ra-226 950
Cs-137 700
Bi-212 Chain Th-232 600
Pa-234 Chain U(235-238) 550
Sc-44 Chain Ti-44 300
Co-60 300
K-40. 200

The most obvious peaks are a big peak made up of the higher count isotopes, then K-40, and then Ti-208, Chain Th 232, 2614 kev. Emerging peaks are from Bi-214, of which there are a few, an SC-44/Co-60 combined? emerging peak, and AC-228. Ti-208 also shows up a little earlier at 583 keV at the noted 1k count.
I'll try to give a reply to your other points tomorrow.