Finite spaces!

So I've been thinking about finite spaces a lot recently. In fact, I think I'm going to do my senior thesis on them (I could probably do a reasonable amount of work and write a decent exposition of Vassiliev invariants, but it just doesn't excite me, and I feel like I lack the background to have much hope of finding new results). Today, I realized that my proof of the fundamental lifting theorem between finite spaces and their continuous models had a flaw. It wasn't too hard to patch up, but it makes the proof more technical and less elegantly simple, which makes me sad. I also for the first time did a little google search on the subject and found it to be well-explored but not hugely popular. The relation between finite spaces and their continuous models was first discovered in 1966, though it seems that the proof was different than mine and did not (directly) use a lifting theorem (I don't actually have access to the paper itself, so I'm just judging from the abstract and the mathscinet review). I also found an example of a finite space that is weak homotopic to a point but not contractible in a paper from 2006, answering one of the main questions I was investigating. I then decided that I don't want to look at any more of what's been done already (at least not anytime soon) and just try to discover more for myself. I found a proof that maps between finite spaces are homotopic iff they are homotopic via a homotopy that only changes at finitely many times (by using spaces of maps between finite spaces to reduce it to the case that the domain is a point).

A question: what method would you people recommend for putting pictures in LaTeX? My pictures will just be simple graphs or drawings of small simplicial complexes.

Some questions I'm currently thinking about:
Is there some characterization of what simplicial complexes are order complexes? It's not hard to reduce this to the question of what graphs are the comparability graph of a poset, but I still have no idea how to answer that.
By the counterexample mentioned above, it is not always true that if two maps between finite spaces are homotopic when lifted to the continuous models, they are homotopic on the finite spaces. When is it true? Can you always subdivide the finite spaces to make it true (I'm certain the answer to this is yes, but I have to work out the details)?
Can we define "singular" homology of a finite space in terms of all maps from chains into the space? Again, I'm sure the answer is yes, and I expect it is a reasonably standard theorem when translated into a statement about simplicial complexes, but I have to work out how to prove it.

Finally, a FLMPotD: Find a finite space that is weak homotopic to a point but not contractible. Hint: start with a 7-point "wedge of two circles", and then add 4 points of rank 2 to obtain a space whose continuous model is a disk but that is itself rather rigid. You can use my theorem mentioned above that any homotopy on finite spaces takes only finitely many "steps" to show that the space is not contractible.