When I did it, I used Zubhium -- they were a web service with a small API that you could install into your app, giving you a mini "app store"-style backend and handling distribution for you. It would host and distribute your APK, connect up to their server when the app launched, check for updates, invalidate old versions, gather crash logs for you, etc. It was very good.
Zubhium are now https://www.vessel.io -- I presume they still have the above features as part of their now-much-bigger service, but when I checked you had to give them a credit card number even to sign up for the free plan, so I've not played with it.
A friend of mine uses http://testflightapp.com for iOS, and it looks like they have an Android version now, so that's certainly worth checking out. A quick search also shows up http://applover.me. @Janusz recommends http://hockeyapp.net/features in his comment.
As @Nanne points out in his answer, the Play Store itself now lets you distribute to limited alpha- and beta-test groups. That looks like it has fairly minimal features compared to the third-party services (no A/B testing, etc.), but will be familiar and free. And it doesn't need an extra SDK rolled into your app.
So, my general answer is that there's more than one professional beta-testing API/service that you could use, that they're generally very useful, quite easy to roll into your app, solve all the problems you're anticipating and more, and often have a free plan to get started. I'd recommend picking one of them rather than trying to roll your own solution.