I can't address the car questions.
For the train tickets: Regional trains, which you will be using for part of your trip and will have the option to use for some other short travel legs (perhaps even all of your trips) if you so choose, there is no advantage to buying tickets early other than not having to stand in line at the train station in Italy. Regionale tickets have one price for each itinerary, and they do not change. You cannot reserve a seat on a regional train, so you can always get on the train you want; however, there's a possibility that you might have to stand for part of the trip at very busy times. (See below.)
The fast trains (Frecciarossa, etc.) offer some promo fares when tickets initially go on sale. As others buy those tickets, only more expensive fares remain. The price difference can be fairly substantial, but Italian trains are not terribly expensive, in general. Seat reservations are mandatory on the Frecce trains; they are included in the fares displayed online.
You can go to the TrenItalia website and check prices for travel tomorrow vs. your actual travel dates to see how much extra you may end up paying by buying the tickets on the spot. Tickets for April have been on sale for a while, so even if you buy tickets today, you may not get the best bargains.
Most people prefer to take a bus between Florence and Siena, and a bus is your only option--other than a rental car--between Civita di Bagnoregio and Orvieto. Bus frequency is sharply reduced on Sundays and holidays.
"Mid-April" suggests that you may be in Italy over the Easter period. That could substantially increase rail traffic and cause crowding on the regional trains. Some of the Frecce might sell out.
I hope someone who has actually been in Italy at that time of year can comment.
Note: I don't know what days around Easter are considered holidays for the purpose of train and bus schedules, beyond Easter itself. It's possible that there will be no bus service at all to Civita di Bagnoregio for several days during the Easter period.