My reasoning for not doing so is that if one were to teach calculus via SIA, one would do so axiomatically and not via the models, just like we teach math majors to reason axiomatically in ZFC before we teach them (if we ever do) how to construct models of ZFC with set-theoretic forcing.

However, for people who are not learning calculus for the first time, learning how SIA works by going through the construction of the models may be the most enlightening way to do it. It could certainly help a classical mathematician who may be suspicious of or not fully understand intuitionistic logic to figure out what exactly is going on.

They’re at:
http://www.math.cornell.edu/~oconnor/sia.pdf

The biggest contrast between Smooth Infinitesimal Analysis and Matt’s system (and non-standard analysis as well) is that while in both Matt’s system and non-standard analysis the object representing reals+infinitesimals is explicitly constructed and can be manipulated directly with the usual classical logic that mathematicians are used to, in Smooth Infinitesimal Analysis, the object representing reals+infinitesimals is presented axiomatically, and you must reason about it using intuitionistic logic in order to make the existence of infinitesimals consistent.

It sounds forbidding, but you can do a lot of pretty neat things with it.

http://www.phdcomics.com/comics/archive.php?comicid=847

I think so; any cycle is equivalent to a B-invariant one, hence to a sum of Schubert cycles. So that handles the groups. Then via Kleiman-Bertini, the product of cycles is going to work the same thought of topologically or intersection-theoretically.