Default [Q] Suggest an Android phone for this specific need

I am looking for a way to download device apps such as the Kreyos phone app ( these are all free apps) from Google Play store and be able to get those apk's off of the phone and on to my Android tablet.

I have been using a Nexus 7 which has worked well but the last couple of devices I have purchased, Google Play will not allow me to download the apk because my Nexus 7 is not a phone. So the question is what would be the recommended pre-owned device I should get that will still allow me to download apps meant for an Android phone but still allow me to get to both Google Play and the apk when downloaded.

I realize it is probably a stupid question but I do not know if any Android phone would work like a Galaxy or only a Nexus would work. Would it have to be unlocked?

If it makes a difference, I use Verizon but if a suitable phone was not available or too expensive, I would be willing to go with a phone I could not use on Verizon but still use Google Play on.

Thanks for any help you can provide.