Can I attach wall studs directly to sleepers on the ceiling joists for an angled climbing wall?
You can attach the joists (studs) directly to the sleepers. The point of using a top plate (not actually a header) in the design you shared is probably to...

Reduce the number of sleepers required
Distribute load more effectively
Provide top edge backing for the wall sheathing

Regarding your second question, you could probably screw the angled studs directly to the vertical studs, but you'd lose any load-bearing benefit from joist hangers, and you'd lose the load-distributing quality in the horizontal ledger/plate. You also eliminate the possibility of putting drywall or sheathing behind your climbing wall without dealing with all those framing penetrations.