- From where did the concept of operator in quantum mechanics came, historically?
This was a gradual development started by Heisenberg's insight. He invented (infinite) matrices (without any prior knowledge of matrix multiplication).
This was followed by by Born, Jordan and Dirac. Dirac's book Principles of quantum mechanics (1930) explains in great detail where the operators come from. Two years later a rigorous mathematical theory of self-adjoint operators needed in quantum mechanics was developed by von Neumann.
- Momentum operator.
Representation of momentum operator comes from the analogy with classical Hamiltonian mechanics, as explained in the answer by ZeroTheHero.
- Kinetic energy
Comes from the general quantization rule: you take a classical expression and substitute the operators in it instead of coordinates and momenta.
EDIT. To address the question asked in a comment to the answer of ZeroTheHero:
There is a book by B. L. van der Waerden, Sources in quantum mechanics, where many papers of the early period are translated into English, with comments.
Unfortunately, Schroedinger's papers are not there. But the main papers by Heisenberg, Born, Jordan and Dirac of 1925-26 (and many earlier papers) are in the book.