In the beginning, Sarah wasn't fond of Batman because he was a vigilante. Later, she still felt that way but only because she felt it made Jim look weak that he had to rely on him. As time went on she revised her feelings towards Batman and saw him as a necessity in protecting the city. From Bruce's point of view, he saw her as a good cop and a good match for Jim. His reaction was part of the beginning of the Bat Jerk phase, where he comes across as cold and arrogant. But he was affected by her loss as he chalks it up as another victim of the Joker.
When Sarah was made Commissioner after Jim had a falling out with Mayor Armand Kroll, she was a bit reluctant, but she contacted Bruce from time to time for help.
And Jim knew that the Joker killed Sarah, because that's in his nature. He would never operate in self-defense. He could claim it to mock Jim, but that's about it.