1. Do you also need a Sanhedrin in order to make a woman eligible for testimony (the conservative midrash you presented)? If not, how is that different?
2. Doesn't the consideration that a Sanhedrin will never arise again practically make that requirement obsolete in some way?

1. In principle, yes. In a place where significant harm is caused, there are mechanisms that could bypass that need. For those, it would be appropriate to have a substantial consensus among halakhic decisors.
2. The same consideration could have led you to conclude that we would never return to the Land of Israel and establish a state. A thousand years ago, or even two hundred years ago, that sounded far more far-fetched than establishing a Sanhedrin.
Beyond that, regardless of future forecasts, I wrote in the past that in my view there is room to discuss whether the rule that "a matter enacted by formal vote requires another formal vote to permit it" still applies after two thousand years without a Sanhedrin. It is pretty clear that those who formulated it never imagined a historical accident on such a catastrophic scale, one that would cause their ruling to freeze Jewish law forever and prevent it from updating and adapting itself. The halakhic decisors do, of course, find workarounds, but there is still a problem here.

What is the minimal definition that, in your view, you would require of a reestablished Sanhedrin for its rulings to be valid for you? What would the procedure need to be?

It's hard for me to determine that in advance. You have to see it and feel it in order to form a position. If right now the entire religious public were to appoint the collection of black-clad people who lead it to be a Sanhedrin, I wouldn't obey. Not for nothing is that something that can't actually happen in practice.