Yes, it seems everyone agrees........ Also find out when they are doing the next Mazda Rev-it-up event, they will show you a thing or two about cornering, and teach you about the 100% rule. I also recommend taking a motorcycle training class.
Get all of your braking and shifting done before the corner if you are giving her the spurs. You need 100% of your traction for cornering, if you are braking then come of that traction is taken away from cornering, and a slight mistake can land you in some serious trouble (or blackberry vines).
In regards to downshifting, there is nothing wrong with it as long as you do it right (and in the right gear), as was said. If all hell is breaking loose in front of you you will thank whoever taught you proper downshifting to get that car to stop quickly and safely. You should always be in some gear anyway, which means no coasting to a stop in neutral, you may have to change your mind in a hurry, and unless you have been driving that car for years you will need to figure out which gear you should be in to get out of whatever you have gotten into.