Take the power to create districts away from the legislature. Give it to a non-partisan commission. 1/3 Democrats, 1/3 Republicans, and 1/3 other/no party. We did that in California, and almost none of the legislators were happy with it, which means it probably did a good job. The state still wound up with mostly Democratic representation, but that's more a product of California being heavily Democrat. I'd imagine that if you did the same thing in Texas, even if the districts were much less gerrymandered, you'd still have mostly Republicans.
Everyone's going to be somewhat political. So no politicians is a good rule, but it still would be good to know the political affiliations of the people doing the districting.
You're objecting to one example of an implementation when you should be focused on the idea. You can clearly define it in such a way as to not rule out multi-party representation as needed. Many other countries have cracked this nut (see Elections Canada). Of course, the history of the US political system seems to be one of NIH...
Sure. I figure if we ever get to the point where there aren't two major parties, it can either be fixed then, or we're under one party autocratic rule and it won't matter anymore.
As possible given what constraints? There's no requirement that they have equal population; it seems like your rule would always require districts to cross zero county lines.
If it could be drawn while crossing zero lines, then yes. But even then it is likely several different options would exist, and each party would try to pick one that favors them.