'Where would they have kept their horses, sleighs, carriages and automobiles?'
I was puzzled about that too when I was in St Petersburg in August 2008. I didn'y manage to get inside, but had a good walk round the exterrior. The palace is built around a courtyard, but this is very small, and while you could get three or four modern cars in there (if you can get through the entrance, which is quite narrow), there is certainly no room for stables. Dimitri was an Olympic rider - he must have kept his horses elsewhere, maybe at the Horse Guards' barracks.