1) What I'm getting at is: Do you or your group usually treat your D&D campaigns as existing within one giant shared universe?
When I am running an official D&D setting, absolutely. It is all one canon.
2) If so, do you use the classic Spelljammer/Planescape approach?
Usually just direct world-to-world astral conduits or portals, sometimes Planescape, rarely Spelljammer.
Planescape and Spelljammer are so detailed and weird that forcing a Planescape or Spelljammer encounter when you're just trying to get from Point A to Point B seems like a disservice to both your campaign and the settings themselves. And I've noticed it tends to irritate players as a result.
When I run Planescape or Spelljammer I generally just run Planescape or Spelljammer. My Planescape campaigns generally only visit weird Prime worlds (like Ortho), rarely the established settings, and my Spelljammer games only use the major cities of the groundling settings as ports of call.
3) Have you taken the next step with a crossover campaign?
I've never gone whole hog, because when it comes down to it I'm just too big a homebrewer. I'd rather be running my own stuff. But I love D&D canon, and will snap up every book I can find that expands it.
For example, our group has used a shared Great Wheel multiverse for 20 years. We've had campaign crossovers. High level characters "graduate" to planar play to keep the homebrew worlds from filling with Elminsters. They team up and have have plane-shifting spelljamming ships and fight threats on an insane scale. The history of D&D and epic level play seems to just beg for this.
This sounds awesome although I don't particularly enjoy high level play. My conception of the multiverse is that this iteration of it is old enough that the paths between civilized areas are reasonably safe, whether those paths cross mountains, atmospheres, or other dimensions. It's only when you veer from these paths that adventure really finds you. So getting from Toril to Oerth is not necessarily in and of itself a quest. It's just getting from here to there. Traders do it, cartographers do it, minstrels do it, low-level PCs can do it.
You cannot get there by spelljammer.... but you might be able to by planar travel.
By the Planescape fluff, you absolutely can, see community of Athasian halflings in Sigil's Hive. And as has been previously stated, Dark Sun fluff is notoriously wishy-washy on the topic.
Could you elaborate on this? I thought Gygax was largely responsible for the Great Wheel.
I think he's drawing a line between the AD&D1 Manual of the Planes and Planescape, which I've seen some other folks do in the past. It's not an unreasonable stance; Planescape is absolutely intended to be the updated Manual for AD&D2, but it's an update that quickly gets weird (and incredibly detailed). The idea that the planes are just another campaign setting really gets under some folks' skin.
Personally, I think it's unreasonable to assume that after thousands of years of Prime Material interference the planes aren't going to look a little like the Prime Material. But if you want a setting where only archmages and Epic adventurers ever see the planes, then yeah, absolutely, Planescape kind of gets in the way.