As a third question we can consider that a completely innocent Oswald was set up in the Tippit shooting by a large scale conspiracy either preplanned or improvised.
How could conspirators know that a totally oblivious Oswald would go to the Texas Theater and watch a movie? He could have gone anywhere or simply stayed on Dealey Plaza.
Yes, only a woman could have kept Oswald out of sight in the TSBD rather than him going out on the street watching the president of the United States, and only a woman could make him go the theater rather than him watching the assassination news in the rooming house. Yet he could have been late, failed to have brought his gun or simply having bragged to friends of his date of a lifetime. No, this would all be too risky.
Improvising creates similar problems, "hey we've got this dead cop here-let's blame it on Oswald, he's nearby". Well, his trip to the theater could simply have been documented by witnesses. Also too risky and you didn't need it in the first place.
Any serious CT should come up with an alternative theory regarding Oswald being totally innocent.