Some thoughts on @friendperidot's musings:
1. I'm in DC, where I don't think moving houses is a thing. However, they've had two spinoffs (Louisiana and Alaska), so I'm guessing it's a thing in places where people buy land without property on it. idk The Snow family has three four generations of house moving/flipping (Paige/Melinda/Kendal/[Sister #4]; Donna/Tony/Gary; H.D.; and maybe H.D.'s daddy?), so it's been a thing for at least that long.
2. I'm fairly certain that most of the attendees for both auctions (pre- and post-renovation) are recruits, passerby, etc.
3. In previous seasons, they showed the staging as part of getting the house ready for auction (along with putting plants around the outside); I can't recall if they still do so. I seem to recall more than one episode where the new owners have asked about keeping some or all of the staging furniture. I imagine the permanent and semi-permanent stuff (eg islands, built-ins, and wacky platform beds) comes with the house.
4. I think that sometimes they have to demo parts of the house due to the original location, which can offer a variety of one-time only issues such as having a lot of trees/power lines or being located in downtown amid a lot of bridges. Other times, they demo a part of the original house that isn't worth the trouble or expense of saving, like exterior brick that makes the house heavier, parts that are built on a slab (and would require them to build a subfloor before the move), or sections that make the building too long or wide to fit on the truck or road. When the movers are planning the renovation, they can take these things into account. For example, the if the flippers have to cut off a roof because it makes the house too tall, they replace it with a lower-profile roof so that it won't be an issue when they move it to its final destination.
I find the whole premise of the show interesting. It makes sense for places where people buy the land first and the actual structure is a secondary consideration. I like the idea of the old houses being renovated and relocated instead of trashed in favor of new builds.