The other answers here are great, but just to more directly answer your questions:
Is there a way that we can prevent lazy opportunists from doing the
same scummy trick to us that they did to Twinone?
You mentioned that they used the Apache 2.0 license. In this case, not much you can do. But what might help is to use the GNU GPLv3 instead, so you're still opening up your code, but you're forcing everyone who takes your code to do the same thing as well. Adding this requirement might be a deterrent, or at least aid in helping the general public become aware that it's a cheap knock-off of something else.
(Optional:) In practice, if you email the Google Play store support
desk, and you ask them to take the fork down — will they?
I think you'd need to contact their legal department. But, to directly answer your question, as long as the knock-off folks complied with your license requirements, then there's nothing you can do. However, by using the GPL, due to the strong copyleft requirements, you're more likely to create a legal case for you to pursue them because there are more requirements to the license, and the odds of them missing one or getting one wrong and thus placing themselves in non-compliance are increased.
I'm not promoting the GPL as a non-commercial license by the way. I'm simply saying that the strong requirements might aide in giving a degree of control in special circumstances. I believe this business model was flawed regardless of license, but others have already touched on this.
At the end of the day, you need to understand that people are going to take your stuff and run with it, especially if you've created something very useful. That's kind of the whole point. Will some people be ungrateful? Of course they will be, but unfortunately I don't know of any legal case where a "you must have good manners" clause was ever enforceable. :)