Police cars are approaching Brewer's store from the East, sirens blazing. A somewhat disheveled That Man (again, hereafter TM) tucks into the vestibule at the store's entrance. He doesn't enter, but just stands there just outside the door and keeps his back turned to the street. Brewer notices that TM is staring (as you like to point out), but he's not staring at the merchandise in the store or the display cases on either side of the vestibule.
How would Brewer know what the man was looking at?
By watching him. A good, old-school salesman is adept at observing what you're paying attention to. That's out of necessity. BTW, it's how you can tell a professional from a dork. A dork will just stand there asking if "you need any help?" A pro will notice what catches your interest, then walk up and open a conversation on the item that's caught your attention.
The police cars make a U-turn at Zang, half a block short of reaching the shoe store, and head back to the East. TM looks over his shoulder, then proceeds in a Westerly direction, away from where the police cars came from.
"Avoiding the police" is just injecting a biased assumption based on what you already believe. Maybe he was looking over his shoulder to avoid knocking down another pedestrian who might be walking by. If the police cars turned around at Zang, why would he need to look for police cars?
Avoiding the police is what Brewer thought TM was doing, or didn't you notice? And when you get down to it, this is all about what JC Brewer thought, not what I think, nor whatever sphincter-clenching notion you'd like to believe today.
TM continues on until he reaches the Texas Theatre, when Brewer sees TM disappear into the recess at the front of the theater. Brewer walks to the theater, doesn't see the guy reappear, and notices TM isn't in front of the theater when Brewer gets there.
You forgot the part where he went back to the shoe store first.
Brewer goes from standing on the sidewalk in front of his store to walking to the doors and finding them locked, then heading to the the theater. That took all of 5-10 seconds. Had That Man walked back out onto the sidewalk, he wouldn't have been able to go far at all before Brewer turned cleared the vestibule and headed West. Brewer would have seen him. Oh, and did I mention that Brewer actually saw That Man in the theater when they turned the lights up?
Brewer realizes that TM had to have entered the theater, and asks the ticket clerk whether she'd sold TM a ticket. He gets a negative response; the guy snuck in.
No, she said she wasn't sure whether he did or not.
This is what she said:
In her 12/4/63 affidavit: "I told him no, I didn't"
In her WC testimony: "I said, 'No; by golly, he didn't'"
What Brewer said:
In his 12/4/63 affidavit: "she replied that she did not think so"
In his WC testimony: "she said no, she hadn't"
To Ian Griggs: "she said no, she hadn't"
To get from that collection of statements to "she said she wasn't sure whether he did or not," you have to pretend most of them don't exist, then take the last remaining one and strip the negative connotation completely out of it. I'm sure that was all just an accident.
the guy snuck in. You might be the only sucker alive who couldn't see that as suspicious and evasive behavior.
Be honest -- you consider it suspicious behavior because you already believe it was Oswald and he was avoiding the police.
Like I said, the question is whether Brewer thought that it was suspicious behavior. He certainly seems to have thought so. And, on any other day, he might not have given That Man another thought. But November 22, 1963 just wasn't any other day in Oak Cliff. I wouldn't be surprised to find a lot of other people calling the DPD that day for things that they otherwise wouldn't have given a second thought to. In Brewer's case, he turned out to be the one guy who really was onto something. And yes, I can see why Brewer decided the That Man was behaving suspiciously.
He was arrested after he struck a policeman and pulled a gun. Let me guess: you don't consider that suspicious behavior, either.
Actually I consider that a false statement. McDonald didn't say he pulled a gun. Besides, he was arrested for murder, not for striking a policeman or pulling a gun.
McDonald did indeed say Oswald pulled a gun:
"Mr. BALL - Which fist did he hit you with?
Mr. McDONALD - His left fist.
Mr. BALL - What happened then?
Mr. McDONALD - Well, whenever he knocked my hat off, any normal reaction was for me to go at him with this hand.
Mr. BALL - Right hand?
Mr. McDONALD - Yes. I went at him with this hand, and I believe I struck him on the face, but I don't know where. And with my hand, that was on his hand over the pistol.
Mr. BALL - Did you feel the pistol?
Mr. McDONALD - Yes, sir.
Mr. BALL - Which hand was--was his right hand or his left hand on the pistol?
Mr. McDONALD - His right hand was on the pistol.
Mr. BALL - And which of your hands?
Mr. McDONALD - My left hand, at this point.
Mr. BALL - And had he withdrawn the pistol
Mr. McDONALD - He was drawing it as I put my hand.
Mr. BALL - From his waist?
Mr. McDONALD - Yes, sir. Brewer, John Gibson, and George Applin all saw a pistol in Oswald's hand during the melee with the cops. How did it get there if he didn't draw it himself? I mean, did a feral revolver that lived in the alley charge into the theater through the open back door then lunge at McDonald's throat before Oswald bravely saved the day by grabbing the rabies-crazed firearm to protect McDonald from it's venomous bite?
The criminal trespass thing would in itself have created probable cause to arrest and search Oswald once Brewer pointed him out.
No, actually it would not have been. They had no probable cause for criminal trespass either. And they tried to search him before they arrested him. No go.
He entered a theater without buying a ticket and sat down in the auditorium. That would most likely constitute criminal trespass.
And the cops didn't even need a warrant or probable cause to stop and frisk you. Remember the NYPD's "stop and frisk" program?
Here's some background from people with an actual legal background:
https://www.legalzoom.com/articles/when-can-the-police-stop-and-frisk-you-on-the-street 
