Found it!
After taking a look at your code, the following CSS rule was blocking the input fields from working:
* { -webkit-user-select: none; }
This is a great rule to have in place, since you don't want users to select text on your page, but it also blocks proper functioning of the input boxes, since they also can't select text. As to why iOS pops up the keyboard just to fool with you, I'm not sure -- because that implies that it something's editable. I guess it's just a bug or something Apple's overlooked. (This also confuses the remote debug functionality of Safari 6 like all get out…)
Anyway, I would try to do something like this in your CSS:
input { -webkit-user-select: text; } (or auto)
and see if it takes. It should, but for my purposes, I just disabled the rule on the entire page. So read the "should" with air-quotes. ;-)
Hope that helps?