I came to a different positions on Romans 13 lately, and one I think actually works. One feature of that is a belief in capital punishment. That doesn't mean I trust our current govt. with it, however.
Regarding Jews and Muslims: Under theonomic law open blasphemy of God (which is admittedly hard to define... I'd define it very conservatively for carefulness sake) is criminal, and trying to persuade others to idolatry is criminal. But simply believing and worshipping a false God is not criminal in and of itself.
I'm not sure how to answer the "quickly" bit. I've been moving in this direction for awhile, but I can't really say why it "clicked" exactly when it did.
Keep in mind that for homosexuality to be a capital crime there would have to be two witnesses. There's a massive problem WRT this with our current State, which actively spies on people and tries to look for crimes to punish (including victimless ones) but that wouldn't happen in a righteous Christian nation. Keep in mind also that a conversion of most people is necessary for theonomy to work and that I support localism. We aren't talking about a global Christian empire built by force here (or even a "Christian-style" America-type empire.)
RE: Rand Paul... I was going to vote for him, but at this point I don't really think I'm going to vote at all. If I do it will definitely be for him. But I'm getting more and more concerned about principles and theology, while practical application is beginning to strike me as boring. Also, if you've seen my threads on RPF, I've gotten a bit less libertarian. You're probably more libertarian than me now
I was mostly teasing. I think you'll get there eventually, but even if not, you're probably 95% there already.
At a certain point you're going to run into arguments that can't be refuted without using anarcho-capitalist reasoning. I lost a bunch of debates in high school because of my logical inconsistencies. Unless/until that happens, I wouldn't worry about it too much.