What card sleeves do you use?
Fantasy Flight Supply sleeves are no longer available. Their replacement, literally, are Game Genic sleeves. These come in gloss (like the originals) and matte, and fit even a bit better than their predecessors.
We use Fantasy Flight Supply sleeves, exclusively. We've tried most sleeves, and are happiest with FFG sleeves by a considerable margin. They're thick, sleek, high quality sleeves.
What foam core/board do you use?
We don't typically use foam core at all anymore, as we're now making our game inserts via 3D printing. However, we still sometimes look to foam core for a simple custom box and other things like that.
We use 3/16" (~5mm) black Elmer's foam board.