Before you make a final decision to rent a car in Barcelona and drop it in Lisbon, check the fees for this (potentially quite high). Unfortunately, while they are next to each other, Spain and Portugal don't actually combine well in one trip. There are a few trains, there are some buses, and there are flights between them. But you have to do the research beforehand to make sure it will work.
Is David correct, and March 12-20 is just for travel between cities? Or is this the total time for the trip? If you have just 10 days for 3 major cities, I'd take the train from Barcelona to Madrid (at 3 hours city center to city center, it's faster and easier than flying, and not too expensive if bought in advance), then fly from Madrid to Lisbon. Not only will this save time and money over driving, but a car is a major impediment in all three cities (particularly Lisbon - just watching the cars was scary).
To buy train tickets in Spain, use the Renfe site, following this TripAdvisor tutorial, and be sure to use PayPal rather than a credit card or your transaction likely won't go through: http://www.tripadvisor.com/Travel-g187514-c80518/Madrid:Spain:Buying.Renfe.Tickets.Online.html
To find flights between Madrid and Lisbon, try SkySkcanner: http://www.skyscanner.com/
As for weather, is 50-60 with chance of rain a problem for you? If so, then it's too cold; if not, it's fine.