There is a minor party called the American Solidarity Party that is a little more of a mix, similar to Christian democratic parties in Europe. They oppose abortion and the death penalty but support universal health care, environmental protections, stricter regulation of financial markets, etc.
Yep, it doesn’t seem to be a common position, for sure.
I have heard of the American Solidarity Party and like some of their viewpoints, but I think that any consenting adults should be able to have a relationship (and the corresponding legal benefits). I also don’t like the religious motivations behind many of the arguments of the ASP; my views against abortion are secular in nature, and I think the religious arguments probably hurt the pro-life movement as a whole.
Political parties like to pretend they have monopolies on certain positions, but that's just wrong. You'll find lots of gay rights-supporting Republicans and a whole mess of pro-gun Democrats. Some voters are rational enough to realize that they should put more weight on the issues that directly effect them than the nice-to-have emotional stuff that doesn't.
Without this, it makes absolutely no difference what the platform is of the Greens, the Libertarians, the Solidarity Party, or any other 3rd party: they can only work as spoilers in an election.
Get us to an electoral system in which the winning D or R candidate is faced with knowing that 40% of their vote came from people whose first choice was the Communo-Anarchist Party, and another 10% from people who preferred the "White Power Now" party, and you're going to see a totally different sort of politics.