Linux Debian codenames
I'll answer your questions out of order: the release team chooses code names (see their task description), two releases ahead; the next three releases are Bullseye (Debian 11), Bookworm (Debian 12), and Trixie (Debian 13); and I don't think we're worried about running out of names yet...
As pointed out by eyoung100, Buster is Andy's dog. As you mention in your updated question, Bullseye is Woody's horse. Bookworm is the intelligent, flashlight-wielding worm toy from Toy Story 3. Trixie is Bonnie’s triceratops from Toy Story 3.
Also, Sid is the name of the next-door kid who breaks all his toys. "Still in development" is a backronym.
You already have a good answer for most of it. But for interest I thought I would play into the math of running out of names:
It has been 19 years since the first codenamed release (Buzz 1996). So far 13 character names are used.
Toy Story has 40ish potential names, assuming no more movies.
If we assume that names continue being used at the same rate -- I suspect they will be used slower, as releases are not happened as often as they once did (image from wikimedia):
But ignoring that, and assuming linear:
13/19=0.68 names used per year. Thus to use 40 names it will take, 40/0.68 = 59 years. Which will bring us to 1996+59=2055.
If we assume that Moore's law continues, (which to be fair, seems unlikely) then the number of transistors per CPU by the time Debian runs out of names will be 2^(59/2) = 759,250,125 times that it was when Debian began using this naming scheme. Or 1,048,576 times that of today. This will be a very different world in computing. Today a high end CPU has 4 billion transistors (like a Xeon), and a super computer might have 10,000 of them (like Tianhe-2), so the supercomputer has 40,000,000,000,000 transistors. An iPhone 6 CPU has 2,000,000,000 transistors. The smartphones of 2055, when Debian runs out of names will have 25x as many transistors as today's best supercomputers. It is hard to say if Debian will still be a functional OS for such computers.
It does however seem highly unlikely that Moore's law will continue up to that point. In which case, computing will likely go somewhere far weirder. Quantum, optical, or other technologies which haven't been conceived of.
In short, Debian does not have to worry about running out of names. The computing world is likely to be so different by the time it does, that such worries make no sense.
Counting from Toy Story to Toy Story 4 (planned to be release 2017), there are less than 8 years between movies, on average. Using @Oxinabox's estimate of less than a release per year, that would require less than 8 new characters per movie (in fact, 22 years/3 movies*0.68 debian releases/year=about 5 new characters per movie), to keep going forever.
I imagine, in practice, if they were to run out (or before), they'd select a new movie franchise.