The barrel that is presently installed is a Navy Luger barrel and has the appropriate and extra high front sight
base for that type gun, not the gun you have just inherited. What that means is that the gun will not shoot to the point of aim, but significantly low...
The only way to get your Luger to shoot to point of aim would be to install a barrel that had a correct front sight base. The only barrel of the ones in your photograph that would be correct is the shortest barrel. The front sight base correctly matches the standard rear sight V-notch that is on both rear toggles that you have pictured. Place the muzzle of the 4 inch barrel against the muzzle of the installed barrel to see this difference in height.
Changing a Luger barrel is not a job for an amateur gunsmith, or even a gunsmith who has not done it before...and I personally do not recommend that you try it.
The Luger frame is easily mis-aligned by the use of improper tools, and can be permanently damaged.
I suggest you read this thread to see what is involved in changing a Luger barrel...
http://forum.lugerforum.com/showthread.php?t=25223
If you decide to install the proper 4 inch barrel, I would recommend you contact member LugerDoc, who has the proper tools and years of experience in barrel changes.
As Bill Lyons wrning above has pointed out,
your Luger would not be legal to attach a shoulder stock, since it is a "parts" Luger and not listed on the Bureau of Alcohol, Tobacco, Firearms and Explosives (BATFE) website as a Curio & Relic.
Attaching any type of shoulder stock to this gun, or any other Luger that is not on the C&R list
BEFORE paying a $200 tax and registering it as a Short Barrled Rifle (SBR) would be a federal offense, punishable by both fine and imprisonment
... so please be careful about what you do with your heirloom...
