Having driven a 3.5t horsebox as my only car for a couple of years after my runabout died, it's not impossible at all, if occasionally awkward to park. I also don't think the "horsey smell" will be at all hard to eliminate, since I assume she's getting rid of the rubber matting that will be over the metal floor. I am wondering a bit just how dirty the previous owners have left it: did they not clean it out at all, before trying to sell it? It shouldn't really be all that stinky if it's been cleaned for sale, at least not to the noses of the horsey people like Alice.
As I know from my mobile-dog-groomer friend, the water and electricity issues are ones that can be addressed. I honestly don't think the idea is all that crazy in itself, assuming she has the support to get the needed repairs done to the box by Jazzer and various mates.
As horseboxes, 3.5t-boxes seem like an attractive proposition, except that very many of them are technically stalled for 2 horses, but would be overweight quickly. Even with 1 horse, many will be pushing it, once you count the driver, the tack, water, feed, and all the other paraphernalia.
I've looked up how much a professional barber chair would weigh, and it seems to be about 25-60kg. That leaves quite a lot of capacity for cupboards, a water tank, lamps, a leisure battery, etc. It seems the industry average female clients per day for a hairdresser is about 5.5 clients. Adding in travel time, reduce that to 4, and at an average water use of 60L per client, Chelsea's van should be ok for water tank capacity too, without going overweight while driving.
At weddings, the biggest problem is going to be time. She won't be doing colouring etc. there, but rather complicated updos and the like, one assumes. So if it's e.g. 5 women in the bridal party, she might need an assistant for the morning. Perhaps the plan is to be able to put a second chair in the box. If the chairs aren't massive, that should just about be possible...