How do I know if my trailer coupler is adjusted correctly?

Sure fire way to adjust it is adjust it tighter until it's hard to close and then back it off just enough that it's easy to close.
And just look under it. You can see how much it engages on the hitch ball.
Grease? Personal preference deal here. If one tows A lot for a long time, same trailer, could actually put some wear n tear on the hitch. Casual user, weekend warrior, will never see any ill effects of not greasing. It certainly doesn't hurt anything though.