I have just shortened my chain by half a link so I can switch between a 16t and 14t cog for road/track useage. For the 16t this moved my rear axle about 12mm forward. I could not believe how much this changed the handling of the bike: It is quite a bit more responsive now.

However, I do have a pretty individual bike with 24" (yes 24, that is not a typo) wheels (see link below). Do you guys get a big change in handling on the 650 and 700 fixies when you change the position of the axle in the dropouts?

A little. I notice a difference switching from bikes with long chainstays to bikes with short chainstays which is effectively what's happening when you move the wheel forward. smaller wheels maybe makes it more noticable. going out on a limb with lots of speculation.

I love your bike btw. I'd really like to build up a folding fixed gear. seems great for travel.

I don't believe that moving your wheel forward a mere 12mm will make a noticable difference in handling. Are you sure its not your imagination?

Shortening the chain by a half link will only move the axle 6mm (supposing the same cog stays on). Switching between a 14 and a 16 causes a 12mm change.
No input on whether that (or 12mm) is noticeable in handling

I am really surprised, but I changed the set up one night, then forgot about it and was ridding the bike all the next day thinking that something was really up with the bike, and then I remembered that I had moved the axle forward.

It might actually be 18mm as I changed from a full link to a half link chain and I probably shortened the chain by 1.5 links. But still, I was really surprised that I noticed the difference.

Cheers, I can not recommend a folding fix enough, I really love it. Airnimals are great, they really didn't make any compromises about frame stiffness or strength so it is a really great folding bike. Also, it looks uber-chic all pimped out for the track, stripped down and with shinny drop bars.