Should we install carpet before the baseboard and door casings?
It makes it easier for the carpet installer because they don't have the baseboards in the way when they're nailing in the tack strips next to the wall.

On the other hand, it makes things a little more difficult to install the baseboard later because the carpet and tack strips are in the way when you're trying to nail the baseboard to the sole plate in the wall.  This is much more of an issue if you're nailing by hand than if you're using a brad nailer.  (I've done it both ways; use a nailer.)  If you're doing a long wall, it can make lining up the pieces of baseboard a little trickier because the tack strip and carpet can get in the way.

On the whole though, I'd follow his advice and use a nailer to install the trim.