I have a paid app on Google Play, and I want to give a copy away for free for someone to try it out and review it. How would I do that? Can I simply take the signed APK and email it to them? Or should I make an unsigned APK? Or, is there a way to let them download it from Google Play without having to pay for it (like some kind of discount code or something)?
Unlike Apple, Google Play does not offer this service (unfortunately to all of us...). You can send them the signed APK as long as you didn't use any licensing service checks that would cause this not to work. This will allow them to test the app, but not to review it. The only way to review an app is to download it through Google Play...
What you can do, however, is have them buy the app, test it, review it and then you can use the Google Checkout console to refund their purchase later...