I am a tax payer in the United States, yet I am not eligible to vote ... Green Card holder!

Why should someone living outside the united states, with no houses in the united states, still pay US taxes, just because they have US citizenship?

This implies taxes are fair, that paying taxes is a requirement for voting, like buying your voice, and that voting would give any benefits in return. Yes, of course it is fair to vote when the vote could indirectly decide laws that make you a criminal in some way or another and no, it is not fair to be taxed, especially if you don't use the majority of infrastructure that taxes are supposedly needed for.

I get where you are coming from, sure there are arguments for it, but it's not like there are none against it, either.

As a UK expat, I don't expect to be able to vote in the UK. I don't live in the UK. Why should I vote there? Why would I feel entitled to?

Why not? Voting was implemented in the USA long before income tax was.

As for the theoretical 2nd house. You would pay property tax on it.