Do ceiling joists need to rest on cross beams?
This is fine.  It's a wooden open web truss system.   

The weight is transferred to the column at the one top point.  The lower bar and diagonal bracing are to prevent the top bar from sagging over a wide span.  The lower bar extends all the way to the wall so there is something for the ceiling drywall to screw into, but isn't supposed to rest on the column.

Here's the more familiar open web steel truss system that is used all the time in commercial buildings. Also called Composite Steel Floor Deck and Concrete Slab