I have a 1936 600 Aero; not on the road but I have ridden one before. A very nice and tractable machine, though not very fast.
Clutch. Like the Velocette clutch, the Douglas flywheel clutch does seem to have a higher chance of being temperamental. For much the same reason, there is little room to spare inside so things need to be in optimal condition.
The best way to check the clutch setting is not at the handlebar, but at the clutch release lever behind the flywheel. With the clutch engaged, this should have the minimal amount of perceptible shake. That will ensure the maximum amount of lift when you pull the clutch lever. As the clutch wears, this gap will decrease, so you need to check the adjustment occasionally. You do not want the clearance to entirely disappear as the thrust bearing will be loaded all of the time and prematurely wear out. The bearing is only used when you lift (disengage) the clutch. Fortunately, the clutch friction material wears very slowly in the Douglas flywheel clutch.
The other issue with the clutch is excessive slop in the components. It is not always wear, as some of it is loose fitting from the factory. The bearings are arranged so they are under the sprocket; more or less. Even so, the chain load tends to tip the clutch plate. A combination of loose fitting parts and minimal lift can allow the periphery of the disk to drag and the clutch does not completely release. Most of this loose fit comes from two locations. The loose roller bearings between the sprocket and the carrier sleeve, and the fit of the carrier sleeve on the hub of the flywheel.
It is difficult to do much with the bearing. The sleeve and sprocket are case hard material. Generally, it is the sprocket bore that is warped slightly out of round. Short of honing and fitting oversized rollers, there is not much that can be done short of a full, engineered replacement of the components. Corrosion and spalling can also take their toll.
A greater source of excess clearance is usually found between the carrier sleeve and the flywheel hub. I have found some of these to be quite loose. As the hub is plated with the rest of the flywheel, I think Douglas left the diameter well undersized so that the sleeve would always fit after plating! This can be a source of problem after re-plating too, as the plating tends to build up a ridge at the end of the hub. I have sorted out this problem by turning the flywheel hub and pressing over it a thin wall bronze sleeve. this also allows the opportunity to lightly hone the inner diameter of the sleeve if it is worn or distorted, and turn the bronze oversized to suit.
Oiling. One drop per second sounds about right; you may find you can cut that back to every few seconds after you get use to it. Do not go by number of turns. Oil viscosity and ambient temperature will many the number of turns mostly meaningless.
Too much oil on the chain sounds like too much crankcase pressure. The oil probably gets dirty very quickly because there is too much combustion gas blowing past the pistons. Could be partial stuck piston rings, or worn rings and cylinder bore.