Problem with downloading files from play market and websites
It is possible that the Play Store is having some issues writing on your phone. However, given that your phone is new(ish) and that there is nothing that's going on a SD card, it makes me think that the Play Store is getting confused.

Depending on what GAPPS package you installed, there could be a variety of tweaks that are done to the package. I would double check with goo.im or XDA to see hat GAPPS to install with your rom/device. Perhaps removing the Play Store, then installing a GAPPS with the right play store app would fix the problem.

Additionally, it could be that you have your runtime on ART instead of Dalvik and the Play Store is glitching up. To check your runtime, go to Settings -> Developer Options -> Select runtime.