Does anyone do that - sort of as an initial step? That is take an admittedly heavier downhill setup into the backcountry, as a way of sort of getting their feet wet?
Yes, it can be done that way. My first backcountry experiences were done hiking with my skis on my back.
And, if so, what is the scenerio? Snowshoe up/ski down? Hike up/ski down? Etc.
Do you already have snowshoes? I haven't done much snowshoeing but a lot of snowboarders start out this way & end up getting a splitboard eventually. Snowshoes can cost as much as a set of skins.
Should I be thinking about going up in mountaineering boots, with crampons, and carrying my ski boots (and, if so, how exactly should I carry those, since most packs I've looked at don't seem designed to carry ski boots internally)?
Crampons aren't really necessary if you are hiking in powder, they can be lifesavers if it is icy. On the other hand, booting in powder really sucks unless there is already an established boot pack.
You can carry your boots in the bindings of the skis, but this doesn't really work if it is very cold or storming because your boots will be frozen when you try & put them on. It can work in the springtime though. There are packs that boots will fit in, I've done it both ways.
Should I be thinking about hiking up in my downhill boots (presumably with the top latches open, for better ankle flexibility), with crampons attached?
You can do this, but your feet may suffer. I usually hike with my boots unbuckled. Crampons are advisable if it is icy.
As I say, I'm looking to take my first steps into the backcountry. After doing the safety part (I'm in New Hampshire, so I'm not too worried about avalanches, but they're here too, so I'll do all the necessary prep there), I won't have enough money (right now) to invest in a touring set up, so I'm interested in any advice you guys have for taking my current downhill setup into the backcountry.
Maybe the best way to start is to get a set of Alpine Trekkers & some skins (still cost you a few bucks). That way you can use your present alpine gear & get a taste of what skinning is all about. It is a clunky way to go, but your initial cost would be kept down.