Gaff definitely isn’t cockney rhyming slang, or even London-specific slang, and it isn’t Irish or Scottish either.
The OED suggests that gaffer, meaning boss, is a contraction of “godfather” (or, less likely, “grandfather”) and was originally used as a term of address for any older man. It gives a few examples of it being used as a title, like “Mr” (e.g. “Gaffer Jones”), which I can’t be bothered typing out.
Gaff meaning abode seems to originate in the 1930s but there is an earlier meaning, a place of public entertainment (origin unknown), from which it might be derived via a slang term for a brothel or a prostitute’s room.
FWIW, gaff takes up a couple of pages and has numerous meanings, some from the same basic etymology (a fish-hook) and others from different sources.
One of the meanings of pad, in the 16th Century, was “a bundle of straw to lie on”. By the early 18th Century, it had come to mean just “bed” and the meaning was soon transferred to the place where the bed was, i.e. the room or abode.
I have never heard “drum” for abode before, though I’ve lived in London for longer than is humanly tolerable. Is it a London thing? Is it in common usage? Is it because of my age?