1. Most cars used in India are pretty small, but upmarket rental agencies have decent selections. You can pretty much forget about sleeping though, as Indian highways are chaotic and noisy beyond belief -- you'll be swerving all over the road to avoid cows, getting blasted by trucks with amped-up klaxons, crashing into giant potholes, etc etc. In fact, I wouldn't recommend travel by night at all.

2. Keeping your driver in Agra for 24 hours is almost certainly worth it. You'll need local transportation anyway, and with your own driver you can avoid the usual souvenir shop detours you'd get from hailing random rickshaws.

3. Yes, there is a 1st class, although it's a pretty far cry from first class anywhere else (except maybe Pakistan). See Seat 61 for a primer. It's infinitely safer than driving, but trains are often delayed hours or even days. The scenery is much nicer than from the highway (well, except for the people pooping next to the tracks), but pretty repetitive -- the stretch between Delhi and Agra is flat, flat, flat.

The Mughal Agra is nice: getting a little long in the tooth in the public areas, but the rooms are pretty good.