For the first question I read a paper today related to Markov Kernels which state it in a more general case (for Markov Kernels)

It has no demonstration at all, but it is a good reference for this basic facts and definitions.

Check it out in section 2.

http://www.sciencedirect.com/science/article/pii/S0167715212004579 The Author is A.G. Nogales, "The existence of regular conditional Probabilities for Markov Kernels".