I recently put a set of Conti GP 4000's on my bike.
The rear tire started collecting cuts and dings almost immediately.
That little voice (you know the one I mean) told me to put one of my Armadillos on the back before I got a tire-ending flat.

So, yesterday morning as I prepared to ride to work I found the rear tire dead flat.
Bummer.

Last night I put an almost new Armadillo on back and this morning I rode to work with the Conti/Armadillo combo.
It felt great.
Super handling and excellent power transfer to the ground.

I have run a similar combo with a Conti 4 Season in front and an Armadillo in back, and I now prefer this type of mixed arrangement over matched tires.
I've never had a front flat, but I've had a front tire slide out from underneath me.
The Contis do the front tire thing beautifully.
In comparison, the rear tire of a bicycle takes the beating, and nothing beats an Armadillo.

As for mixing sizes, I don't know.

That said, putting a lightweight, good handling tire in front, and a tough, heavy-duty tire in back makes sense to me.