Code humor and inclusiveness

Practically? Possibly.

It appears the etymology's independent. Having poked around on en.wiktionary.org:

The "gay man" usage is a contraction of "faggot", which as well as meaning "bundle of sticks", "bundle of metal rods", "burning ember", and "English meatball" also came to mean "shrewish woman" and later (I dare say by extension of the "shrewish woman" sense, with some possible influence from a Yiddish term "feygele" which means "little bird", "loved one", and (offensively) "homosexual man") came to mean "homosexual and/or effeminate man".

The "cigarette" usage appears to me to be likely to be a back-construction from using "fag-end" (a very old term meaning "remnant or worst portion", etymologically distinct from the above) to refer to the unsmoked remnant of a cigarette.

No, the word as used most commonly in literature before the cigarette slang, meant junior boarding school boys who had chores for seniors. All intended to be character building and young gentleman would never abuse such power, would they?
You can contort any words for cigarette to be homophobic following your argument, which disproves it by reductio ad absurdum.