That is a military pistol. The problem is that the Erfurt toggle does not match the receiver or barrel. Erfurt was a government (originally an Imperial Prussian) arsenal; it was the site of one of the only three original installations of Luger tooling, the others being the DWM factory and the Swiss arsenal at Bern.
The three markings are millitary inspection marks, applied to the pistol before final finish and bluing. They signify inspection and approval of 1) the receiver alone, 2) the barrelled receiver, and 3) the entire pistol. The letters are the initials of the inspectors, but their identity has long been lost. The eagle is the Reichsadler or Imperial Eagle, signifying that the pistol had undergone final proof and test firing and was accepted for service.
The gun has parts from at least two pistols; those with "44" are probably from the gun numbered 1644, which seems to be the number on the grip frame (there should be a following letter; the other parts are from a gun with a serial number having "43" as the last two digits.
I wish you luck in finding parts numbered "44" to replace the "43" parts (no, the factory did not make a mistake, though some German soldiers might have). At one time, some of the Luger collectors tried to set up a "parts exchange", but I don't know if they were successful. The mismatched magazine is common; matching magazines are desireable and add to a gun's value, but are not worth worrying a lot about.
Do not try to "touch it up" or have it reblued. That would cut the value in half.
The gun is in good shape, and while mismatched, would likely bring close to the $1000 mentioned, at least with the holster, which would bring $100+ by itself.
Do NOT store the gun in the holster or gun "rug". Storage is best in a box, in a dry area; if in a safe, use a "golden rod" or similar dehumidifier. Boxes can be obtained from Roberts Arms Specialties, email@example.com
or 304-728-4472. Some folks store guns in zip lock baggies, which are inert and afford a degree of protection, then in a box. In any case oil the gun well with a good gun oil or something like G96 Gun Treatment before storing.