-- Why didn't Oswald go directly to the movies (Texas Theater) after leaving the TSBD rather than going home first?
Obvious answer based on his actual actions is that he wanted to get changed out of dusty\dirty work clothes into more casual clothes. He went to the boarding house got changed and then went out, seems like a normal set of actions.
-- Why did Oswald take a revolver to the movies (Texas Theater)?
LN Answer - As the shooter, he wanted to use it to avoid arrest, like he did at the Tippet scene.
CT Answer - 1) He had worked out that he had been set up, and was scared or 2) He was just being a lad and carrying a gun made him feel more macho. or 3) It wasn't his
I've always wondered why not take it in the morning, when he was going to do the actual shooting.