You can use libgdx and is Java. Though you can make desktop games some people use it just for android development. There is slick2d for java but the creator now uses libgdx. Lots of people said it's a fantastic framework. There is unity free which is good but you will probably need some 2d plugins. Then there is c# and xna which c# is somewhat similar to java in which you can learn c# pretty quickly. There is lua with love2d , Giderios. Lua is easy and is fun to use.
I certainly think it is possible, I think Sony will probably charge to host your game on Gaikai's cloud services. I think it is best to use the hardware. The public SDK has to use a safe language like C#. If you were allowed to c++ like the big studios do then people can do harmful stuff to the ps4's security. Big studios get a ps4 dev kit to dig deep and use it's hardware to the max with c++. While we get a SDK that is nothing like the big studio's use and c#. Which is good enough. If Microsoft was able to do indie game development on the 360. I am sure that ps4 can do indie game development. It would make me very pleased to see Sony pull this off. Also unity and Paradox3d(both c# engines, but paradox3d is in the works) can deploy to next gen consoles.