...which is how I came to ask the question, Who Is Andy's Father?
You'll note that there aren't any pictures of him in the house. Don't believe? Go look. I can wait.
See? Lots of pictures of Andy and Molly, individually and together, and one or two of Andy's mom, Mrs. Davis, but none of Andy's dad. Why is that, do you think?
The obvious answer is that Andy's mom took them down. Why would she do that?
Let's think this through. Andy's father never appears in any of the films. He's never even mentioned. If Andy's mom was still married to "Mr. Davis," that wouldn't be the case. He's deliberately absent from the films because he's not part of the family any more.
If he had died, there would still be pictures of him in the house. Divorce is a possibility, but I don't believe Andy's mom would have removed all the photos of Andy's father just because of the divorce, not least because of how Andy might react to that. If there was a divorce, then Andy's mom won custody, and while Andy is clearly happy with his mom, it'd only be natural for Andy to want something of his father, if only a picture.
The remaining alternative is that Andy's father abandoned Andy's mom, and Andy, and Molly. This would explain why Andy's mom removed all the pictures of Andy's father. It would also explain why Andy's mom is moving the family to a smaller house during Toy Story 1. Go ahead and compare the house at the beginning of the film with the one in the final scene--post-move house is smaller. it also explains why Andy's mom is bringing so little furniture to the new house--take a look at the contents of the moving van in TS1, there's practically no furniture in it, and the time from when the movers arrive in TS1 to when they leave can't be much more than an hour. Not only is the new house smaller and needs less furniture than the old house, but Andy's mom, having been abandoned by the former Mr. Davis, is in all probability facing some difficult financial times and is selling not just the house but some of the furniture.
But the matter of "Mr. Davis" doesn't end there. Consider Woody. We know from Toy Story 2 that he's a valuable and rare collectible. How did he end up in Andy's hands?
(The formidable James Wallis, creator of the wonderful, sui generis roleplaying game The Extraordinary Adventures of Baron Munchhausen, recently wrote about this on his blog. I had this essay in mind long before Mr. Wallis wrote his piece, but what he wrote--which was, erm, far from his best work, shall we say--impelled me to finally set this all down).
According to Andy's mom, Woody is an old family toy. But if that's really the case, why is he in mint condition? If he's been played with before, by a previous generation of Davis children, why does he look like he's fresh out of the box? And why does he act like Andy is his first owner?
We have to assume that Woody was, indeed, an old Davis family toy. The logical assumption is that he was Mr. Davis' before he was Andy's. (We could extrapolate other ways in which Woody joined the Davis family, but Woody being Mr. Davis' toy is the least unlikely). But how did Woody remain mint if Mr. Davis played with him? And if Mr. Davis played with him, why doesn't Woody remember him? We know the other toys remember their previous owners--Jessie remembers Emily, Lots-O-Huggin' Bear remembers Daisy--why doesn't Woody remember Mr. Davis?
If--as I think is a reasonable assumption--Mr. Davis abandoned Mrs. Davis, Andy, and Molly, he did so recently. Molly's only a year old, and it's probable that she is Mr. Davis' daughter, which means that Mr. Davis abandoned Mrs. Davis when she was pregnant or had a one year old to raise. Which ain't cool, at all, and hints that Mr. Davis might be worse than just a deadbeat dad, he might be a genuinely Bad guy.
There's also the matter of Sid Philips, the psycho boy next door in Toy Story 1. Sid is disturbed in the ways that Andy is not...but they both have vivid imaginations when it comes to toys, both are resilient (notice how quickly Sid recovers from the disappointment of the sudden rainstorm in TS1), both are energetic, and while Sid is cruel to his toys he loves his dog Scud.
Could Sid and Andy be...related? Sid looks nothing like Andy, but then, Andy looks nothing like Molly. During TS1 Sid is 10 years old and Andy is 6. Did Mr. Davis abandon Mrs. Davis because Mrs. Davis discovered Mr. Davis' affair, previous or ongoing, with Mrs. Phillips?
Now, as we see in TS1, when toys are tortured, they are traumatized. Only one of Sid's toys can communicate in any way, and none of them have the ability to make facial expressions, whereas most of Andy's toys can talk and move their faces. In the world of Toy Story, as in ours, torture leaves a permanent mark on its victims.
But there are many different types of trauma, and some are more subtle than others.
Consider Woody's relationship with Andy. Woody is devoted to Andy. Fixated on him, really. Is willing to do anything to get back to him (as we see in TS2 & TS3), refuses to admit that Andy has outgrown him (in TS3)...I'm not sure that "obsession" is too strong a word for Woody's devotion to Andy. Woody's feelings for Andy are clearly much stronger, much more obsessive, than what any of Andy's other toys feel for Andy.
Why? Why does Woody not seem to have any memories of his previous owners, but is fixated on his current one?
Because his previous owner did something to cause some trauma in Woody, I think, a trauma which gave Woody either traumatic amnesia or a deeply-embedded case of denial of memories as well as a case of idealization (in the psychological, defense-mechanism sense) of Andy. Woody can't remember (doesn't want to remember) his previous owner, but his current owner, Andy, is wonderful, worth following to infinity and beyond (idealized owner, safe owner, everything's okay with Andy, everything's fine).
Obviously Woody was never tortured by his previous owner--Woody couldn't have been, not and be in mint condition. But he could have witnessed torture. That would certainly qualify as traumatic.
Woody is fixated on Andy. But he actually considered abandoning him for a new life with Jessie, Bullseye, and Stinky Pete. It wasn't the lure of the museum which temporarily changed Woody's mind--it couldn't have been. No, it must have been the prospect of a reunion with Woody's toy family.
Reunion? Yes, reunion. Who do you think Woody saw being tortured by his first owner?
I'm sure there were other toys that Woody's first owner tortured. But I think it's safe to suppose that the rest of the Round-Up Gang were some of the victims there. The reason Woody escaped the torture? Woody's previous owner just never got to him--moved on to other things, and left Woody in the attic, in the dark, wondering when he was next to go under the magnifying glass or scissors...in the dark for years, for decades, until he is given to Andy, wonderful, kind Andy.
And so the obsession is created.
I've painted a dark picture of Mr. Davis, Woody's original owner: abandoner of wife and children, possible adulterer, torturer of toys. This begs the question: who is Mr. Davis?
Andy was born in 1989 (6 in 1995, the year of TS1). In TS1 I'd put Mrs. Davis at around 40 (which is how old Laurie Metcalf, the voice of Mrs. Davis, was in 1995), and we can assume that Mr. Davis is roughly the same age, meaning he was born in the early/mid-1950s. Woody's tv show lasted from 1946 to 1957, so--assuming Mr. Davis was given Woody & the Round-Up Gang in the last year possible, and when Mr. Davis was young--we can put Mr. Davis' birth year in the 1951-1953 range, which would have made him 35-37 when Andy was born.
Mr. Davis: born in 1951-1953, a torturer of toys, a man who abandons his wife and children. We have the makings of a true villain in Mr. Davis. So who was he? He would have been active in his twenties during the 1970s and still alive, though temporarily settled to a suburban lifestyle in his forties.
Who fits this profile? What villain of tv, movie, or comics could this apply to?
Edit: I love Marcus Rowland's suggestion: Dexter Morgan!