For your #3, maybe I'm missing something, but don't sentences like
these have noun clauses as objects of verbals?
Believing THAT THEY HAD CROSSED THE MISSISSIPPI, the
travelers rejoiced at having commpleted half their journey.
Not knowing WHAT LAY AHEAD OF THEM contributed greatly to their fear.
Gamblers need to know WHEN TO FOLD UP, WHEN TO WALK AWAY...
Is this the sort of thing you're looking for?