As far as size, worker bees can, and do, squeeze through 1/6" square holes. If it's wider they can even get through 1/7". One bent wire on #7 hardware cloth lets them in.
As far as dead bees. Bees at feeders die. Sometimes they drown. Sometimes they fill up and can't seem to fly. Maybe they're old. I don't know why, but they do die. If it's only a few, don't worry about it. If it's a few dozen, that's still not outrageous. If it's a few hundred I'd be looking for a different system.