1. Unfortunately, I'm not too sure. The ascent looks like it shouldn't be a problem. Descent looks steeper but probably doable? I'd expect some snow on the descent though. There was a good amount just on our descent of Hailey Pass. That's the pass on the left.
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2. You could definitely get there in either vehicle. It has some good pot holes and big rocks sticking out, but I always think about whether my Civic could make it up the roads I drive to trailheads, and I was pretty confident I could do it if I drove it carefully. So a minivan should be fine. Just drive aware because some of those potholes, hit at high speed, could be a bad deal.
3. You could bring microspikes just in case, since those north-facing ascents on your route will likely have some snow still. I didn't use mine when we were on snow though because the snow seemed to be at the point where it would just ball up on the spikes and make walking harder. We just used trekking poles for stability and potential self-arrest and no one ended up sliding anyway. There's definitely going to be a lot less snow now because of the warm temperatures we've had all over the past week, and it'll still be warm this coming week. So if it were me, I'd skip the axe and pack the spikes. But I'm also not terribly experienced or well-versed in that and someone else could probably give you more certain advice.