Alvin,
The chamber cut on a standard-barrelled DWM indicates that the gun was originally an Artillery, and has been rebarrelled. The chamber cut was mandated on Erfurt P-08 after 1916, which oddly enough only made LP-08s in 1914.
Ken,
Thanks for the addtional pictures. The c/N proofs were applied by a state proof house, probably in Suhl or Zella-Mehlis, when the gun was re-proofed because the barrel was replaced. The photo of the under-barrel marks are too indistinct to make out, are you sure the number is not 8,87 ?
The death's head mark, and any purported connection with the SS, is controversial and not firmly established--there are competing theories. Oral history capture reports are notoriously unreliable; if your Grandfather's story could be verified it would be very useful. You will probably find some material if you do a Forum search on "Death's Head" and "Totenkopf". You may find more in-depth discussion on this topic at Jan Still's Luger Forum,
http://luger.gunboards.com.
--Dwight