It's graded county road, but the end of the road that goes past Holy Cross goes to Homestake Reservoir, which is pretty cool. There are a few other dirt-road spots in around Holy Cross that are cool to just poke around exploring as well.
Mosquito Pass could probably be done with careful driving in a Subaru, but it's pretty cool. The Matchless Mine is on the Leadville side and if you're history-geekly, it's kind of cool to check out. There are a couple other mine sites on the way up Mosquito that you can look at from the road, but can't look into (private property).
You can theoretically go, on dirt, from Camp Hale to the Vail Pass rest area, I've done most of it, and it's mostly not-hard trail, I can't say 100% it goes through, but I think it does. It goes out the northeastern end of Camp Hale, winding around to the east side of Red Cliff, then over Shrine Pass. I wouldn't want to take a crew/longbed that way, but I think a stock YJ with a decent driver could do it. Some incredible views along the way, and more than a bit of old Colorado mountain history as well.