How can I build a social game like FarmVille or Social Empires in 2 or 3 weeks? What tools can help?

First, you need great people that are already working together, to hit the ground running. A group of people that have never worked together will spend a lot of time running into walls. Second, you need great middleware, so you aren't spending your time on (re)inventing technology, instead of creating value for your customer. You can start with something like Unity and their library. Also, 2-3 weeks won't get you far. Don't buy into the myths that these things happened in 2 weeks. They were measured in weeks to reach the minimum viable product but 2 weeks is more in the realm of a validated prototype.

You can't.  A game like Farmville (or the 2014) equivalent will probably have a team of 20-30 engineers working a year. If they could do it in 2-3 weeks, they would.  You can probably build a very basic prototype in 2-3 weeks, I would say use Unity to do it or even Epic's Unreal engine, but to build a competitive game will take a lot longer.