Origin of 'skinny' uncertain. Noted in print in 1959 in an article on slang, but probably WW2 or earlier. Perhaps originally meant a 'brief summary' or the 'naked truth'. Appeared in "Gentleman's Quarterly," but I don't have the date.
Also used to mean "What's Up?"
(Oxford English Dictionary)
Sadly, vol. 3 of J. E. Lighter's "Historical Dictionary of American Slang," entries after 'O', has not yet been printed.

Low down- Probably first used in lower class slang, U. S.
1907- Robinson "Comics," "The Lowdown Kid."
1908, Fisher, "A. Mutt." "I can give you the low down on A. Mutt."
1915, T. A. Dorgon- "Aw, give us the low down on them, Bill."
(Incorrect attribution in the OED).
1920, "Colliers Mag." - "He calls me back and in about twenty minutes I have the low down on Monsieur Kane Halliday."
By 1924, P. G. Wodehouse (English) used it in one of his novels.
References from J. E. Lighter, "Historical Dictionary of American Slang, vol, II."
Low down (now lowdown) in common usage before WW1. Not originally a military term.