That is one thing I miss from my old HTC phones. In the mail app, it gave me a choice to ignore contact/sync info from different accounts. So I was able to tell it to ignore Google contacts without having to delete the Google account. I have not found that functionality on the S4.
Now mind you, there are very few things I miss about my HTC. Love my S4. But that was a nice feature.
EDIT: As far as downloading outside of the Play Store, some apps to have developer websites that allow that. You can directly download the APK, put it on your phone's internal memory and then install it. But not many do that. Most of the time, the websites just send you to the Play Store.