Robert,
Welcome to the marque! The engine number should be on the crankcase, just above the base of the rear cylinder. The frame number may be 5927 with the 'K' the remnant of the superimposed DK trademark that Douglas used. If so it would be a 1912 frame. That would be early though not impossible. More likely you are missing a numeral to make up a five digit serial number and it as a WW1 bike or later. The transmission would date from 1921-ish onwards, when they started using the 'LG' prefix. Douglas made the 2-speed clutch-less model right up to the end of 2-3/4hp production in 1926.
The hand pump feeds to the oil sight glass (dripper) and then enters the engine at the base of the front cylinder via a one way valve. There is no automatic pump. You pull the hand pump plunger all the way out, and it returns slowly via a compression spring, forcing oil through the dripper at a rate determined by the needle valve on the sight glass body. Ideally at a rate to keep the engine well lubricated, without introducing too much oil and overpowering the piston rings and just making a lot of smoke.
-Doug