There's a bit of brilliance to figuring out the various toys' personalities, except within the text, it's quite limiting for who they get to be. They don't fully have free will, is what it is.
Mr. Potato Head is for "Ages 3 and Up" so he's a bit pretentious, full of himself. He's too good for, well, Andy's sister; that's probably the only kid under 3 close by. (He's also both the most expressive and the most... incomplete because of his interchangeable parts.) Rex is not an older kid's dinosaur; he's made for toddlers so he's timid, not even remotely as scary as he wants to be. Slinky Dog can lengthen his body, so he is both literally and figuratively stretched thin by his position as Woody's de facto assistant. Ham is not even a toy; he's made for holding money, but that makes him stingy, selfish, a bit of a know it all because he's not like the rest of them. Mr. Spell is made for teaching so, of course, he arranged the "plastic erosion awareness meeting." The three-eyed aliens (in this film) are stuck in a claw machine so they worship the claw that can take them away at any time--though realistically, it probably takes no one... ever. They are also trapped in a confined space with identical aliens, so they lack individual definition; they lack identity. They are like brainwashed cultists, stuck on the singular idea that their god is a claw hanging over their heads.
Woody is a sheriff toy--and, for now, the only cowboy toy, so he is a) isolated and b) prone to taking charge. Because Andy is clearly into cowboys, Woody often plays the leadership role during playtime as well, so of course he needs to be controlling. And, he is threatened by the presence of an alpha male toy. Woody is also a strange contradiction in that he should be a tough guy cowboy (keep in mind, we have not yet seen his portrayal in Woody's Round Up), but he is made of cloth, a little too flexible and gangly and, dare I say it, maybe a little effeminate. And prone to emotional outbursts. Before Joss Whedon got involved with the script, Woody was "a bitter toy who berated and insulted all the other toys and was bound and determined to destroy Buzz."
(Woody is voiced by Tom Hanks, who once starred in in Bosom Buddies, in which he regularly dressed as a woman. Buzz, on the other hand, is voices by Tim Allen, who once starred as the overly confident and overtly male Tim "The Tool Man" Taylor on Home Improvement. (Sid has a Binford Tools toolbox, by the way.) I almost want Woody to put on a falsetto voice at some point and ask "Who is it?" and Buzz to grunt like an ape.)
Buzz is quite masculine, on the other hand, with thick, presumably muscular limbs and (very) wide shoulders. Buzz is a space ranger. He's used to getting things done. That's his character. He's also a bit trigger happy, so you gotta wonder how kid friendly his show is.
(Speaking of Buzz being trigger happy, could it mean something that Buzz aims between Woody's eyes, the exact spot that Sid will later burn? Is Sid just as stuck being his character as Buzz is his? ...more on this below, maybe.)
Buzz immediately starts getting the other toys to work out, to be better versions of themselves. He gets Rex to roar like a real dinosaur, even. Woody is okay with everyone being who they are. Buzz wants things to get better. He wants to repair them like he wants to repair his ship.
Woody, because of the "rules" is nearly run over and killed by a semi-truck. Buzz, in that same moment, because he is still deluded about being a real space ranger, runs out of the way and hides. It isn't that Buzz thinks he's a real guy, it's that he thinks he has a choice, that he has free will. The claw and the three-eyed aliens are like a symbol of the entire plot. Buzz has been selected by Andy, but that means he must lose his illusions of freedom (like the alien Sid takes home must immediately learn a hard lesson about what life is really like on the outside, as a chew toy for Scud).
And, what of Sid's freaks?
Like Frankenstein's monster--and that's the second time this month that character has come up--Sid's toys are a mishmash of parts. What does that make of who they are? The one with the fishing pole doesn't even have a head anymore. Where is it's brain? For that matter, toys don't have brains in their heads, so how do they think? How do they make decisions? Or are they just toys for the Pixar animators when they aren't toys for Andy or Sid?
For that matter, isn't Sid just another toy for the animators? Toy Story 3 suggests that Sid turns out okay, but here he's a bit sadistic, perhaps evil but more likely a bit misunderstood. He's a kid. The movie may frame shots in his room like it's a horror film, and that lightning shot like it is a Frankenstein film, but you gotta remember, he's just a kid. Woody and Buzz, on the other hand--they are adults. That is the way they are made. Yet they turn on this kid and scare the crap out of him, when really, if they could get all the toys to team up--and break the rules--they could have just run away rather than getting revenge.
I once wrote in this blog about a character in a screenplay I wrote who figured out she was just a character and she tears down the movie around her and leaves. I followed up that description with the following:
I say all that to point out that I take fiction seriously. And, I understand that inside each story there should be a reason for suffering, a reason for pain, and a motivation for what characters do in response. But, I also believe that fundamentally, a lot of what characters go through comes down to choices made by a writer typing away like I'm doing now.
I kind of feel sorry for Sid is where this is going. Sure, Buzz is put through the emotional wringer, but he's got other toys around to support him afterward. Who does Sid have? He has probably antagonized everyone he knows. And, I doubt his father is much help--note how Scud seems to be afraid to go into the room with him.
There's a great theory online about Sid and his appearance as a garbageman in Toy Story 3. He knows that toys are alive so he got a job dealing in garbage to save all the damaged toys that people throw away. I didn't give Buzz the Christ-Figure point for 15.75 atonement yesterday, but really his actions and his near death by rocket does atone for all of Sid's sins. Sid has been mistreating his toys and he learns to do better. (It just takes another two movies to figure this out.)