1. It's the olde English terminology for a bundle of sticks. I doubt anybody really uses the word for that meaning anymore. Maybe some old people in a corner of cornwall do, but they don't count.
2. A term used for a disgusting British delicacy often eaten with peas and purchased from a low rent fish and chip shop. It's basically the ass and innards of a pig ground up into blobs of indistinguisable brown meat. It is NOT a british term for a cigarette. If you went up to a guy on the streets of london and said 'have a faggot?' they'd probably beat you up. The correct word in that situation is 'fag'.
3. The most common usage today is for a homosexual. Who knows how it evolved into that but with the older meanings about sticks and balls of ground meat you can probably guess. The word has been reclaimed(in the same way as the word 'gay') by teenage boys who feel the constant need to affirm their masculinity by putting down gay people and femeninity.
"You faggot!" - bored teenage youtuber
"I'm not a homosexual, not that I am against them. Please, shut up.' - rational person
A word obviously disliked by most of the people on urban dictionary for being 'politically correct'. While that may be true, it is no less 'proper english' than the use of the word black for people who are usually varying shades of brown and tan.
If the word is so detestable maybe we shouldn't use hypenated-american terms at all. Should we refer to chinese americans as yellows now? Or JFK as the first redhead catholic president?
The truth is that african-american is suited to some situations and black is suited to other situations. Just like caucasian americans are routinely refered to both as white and by terms like 'english-american'. Just because africans were brought on ships hundreds of years ago doesn't make that heritage any less important to a black person as european heritage to a white person.
"Look at me. I'm black. Look at my black skin" - Colin Powell
African is a useful term for implying geographical heritage. There are people in South East Asia who for all purposes look like what we know as 'black people'. When refering to the geographical origin of the american black population african-american is a useful term.