1 - He didn't deny it to the police. He only denied it to the press. Just days before the assassination Kennedy was on the radio making a particularly anti-Cuba speech, which upset Oswald.
2 - The day he assassinated the president, he left all of his cash and his wedding ring in the home his wife was staying, after unexpectedly stopping by the night before to pick up his "curtain rods." When the police arrived at her home, she led them to the garage to show them that his rifle was kept in the garage and lo and behold, it was gone. She herself testified that at that moment she knew he had done it.
3 - I don't think he was in jeopardy of losing his job, but he suddenly had an opportunity to do something "great."
4 - He asked for the ACLU attorney because he knew that attorney also represented the communist party. https://en.wikipedia.org/wiki/John_Abt