Learn English – Meaning of “Green” and “GreenBack” in American english

I'd say there really isn't much difference in usage between American English and British English. In both dialects, it can be used to mean "it's alright", and dismissively to mean something like "shut up, I'm annoyed". A lot of it depends on tone of voice.

1. Did "long time no see" arrive in U.S. English from forms of pidgin English spoken separately by both some Native Americans and some Chinese immigrants?

The earliest recorded examples are from native Americans, but it's plausible it was used in other types of pidgin English at the same time.

2. When did this turn of phrase first gain the recorded notice of an American English-speaking author?

It has been recorded by American English-speaking writers in 1900. The author Raymond Chandler used it in a 1939 newspaper and 1940 book.

3. When did the phrase cross over into use by native U.S. English speakers among themselves?

Chandler presumably helped popularise it with detective stories and film noir of the early forties.

The OED says it's a "Colloq. phr. (orig. U.S.) long time no see, a joc. imitation of broken English, used as a greeting after prolonged separation."

Their earliest quotation is 1900 from a native American:

1900 W. F. Drannan Thirty-one Years on Plains (1901) xxxvii. 515 When we rode up to him [sc. an American Indian] he said: ‘Good mornin. Long time no see you.’

Their next quotation of 1939 shows it was fully naturalised:

1939 R. Chandler in Sat. Evening Post 14 Oct. 72/4 Hi, Tony. Long time no see.

Their next is also from Chandler, in 1940's Farewell, my Lovely.