I did this in January 2014 and I'll try to address your questions:
1) Renting a car in El Calafate is no problem. After some googeling and reading reviews in the Internet, I decided to opt for
Nunatak Rent a Car and never regretted the decision. They reliably picked us up from the Airport, even though our flight was a couple of hours late and in generally everything worked without a hassle. They also provided us with a map of the region and told us where we should refuel since petrol stations are quite rare in this area.
2) Driving to Torres del Paine was also not a problem. Just leave El Calafate on the eastern side of the city and follow the road towards the Airport. Pass the airport and follow then Highway 40 to La Esperanza. A couple of kilometers before Cancha Carrera, you will pass Tapi Aike. Stop there at the petrol station and refuel your car. That's the last petrol station before the border and there won't be any petrol station in the national park and also not between the Chilean border and the national park. At the border crossing in Cancha Carrera (be aware, Google Maps does not know this road) leave the highway and follow an unmarked dirt road for a couple of kilometers. Finally, you will reach some buildings that look like farms, but that is actually the border crossing. Stop there, go into one of the buildings with your passport and the car insurance card. The whole procedure just took a couple of minutes, even though they couldn't find our passports on their obscure list of countries in the world. However, after another few kilometers, you reach the Chilean border. Directly after the Chilean border, there is a restaurant and souvenir shop. Sometimes they have gas there, so ask for José, the chef, and see if he can sell you some gas. Then depending where on the national park you want to go, there are various options, but probably, just follow the road Y-200. Be aware that there is no petrol station in the national park.
3) At most head trails, there are big parking spaces where you could park your car for a couple of days. I think it should be a problem at all. However, I would maybe thinking about putting a visible note in the car with your expected return date, so that park rangers are not going to search for you when they notice that the car is there for several days.
4) Same procedure as 2, but in the opposite direction.