No reason you can't use the Kodachrome with flash, but the combination probably isn't the best for an ad hoc learning experience.
Probably best to rewind the kodachrome manually, if the camera allows it, stop turning as soon as you feel it snag a bit as it gets to the end of the leader on the winding spool. Then mark the leader with how many exposures are on it and reload it when it's a more appropriate film.

Much depends on what sort of things you're doing with the flash, and what sort of flash, i.e. TTL, automatic, completely manual, etc.

Otherwise, using flash with Kodachrome is no different than using it with any other film. Since slide films have less latitude than neg films, shadow detail might be a problem, but it all depends on what you're doing.