Did Tars Built the Tesseract?

Tars the robot built the Tesseract. He did so in order to survive. At the end of the movie Tars is fully "Strong Artificial Intellegence" meaning he is at a point where his intellectual capability is a functional equalavent to a human and is also capable of recursive self-improvement.

How did TARS get the quantum data?

Finally, he takes the quantum data that TARS has collected from the black hole's singularity – data which could only be observed from within the black hole – and sends it, via morse code, to the second hand of Murph's watch. It's this quantum data which solves Brand's gravity equation.

Who had built the Tesseract for Cooper and had been communicating with him?

Rather than attempting to use computer graphic animation throughout, for many of the special effects Nolan had large elaborate sets built. Here we see Cooper attempting to communicate with Murphy through the Tesseract, using the wall of books in her room.