If the quadrant is rubbing on the bottom of the frame, it will have taken the black paint off the quadrant. Can't see from your photo as it's too dark. Get some light in there!
You definitely do not want to pour engine oil into the steering cable conduits. The Lewmar recommendation is PTFE grease, I've had good success with Super-Lube, which is a silicone grease with PTFE and sure not to attack the plastic of the conduit (unlike motor oil or WD-40).
The quadrant shouldn't be touching the frame, so there is no need to grease there - if it is rubbing, adjust quadrant height (top nut/washer or quadrant height) until it's not touching anymore.
As it sounds like your rudder is at least somewhat difficult to move without the cables connected, you'll have to investigate that direction. Some people on here had success with pouring vegetable oils down the rudder tube to lubricate the bottom bearing