etymology - A star is born!

For your (1), regarding the use of 'star', the earliest uses I could unearth of 'star' in the figurative sense,

5. fig. A person of brilliant reputation or talents.
   a. An actor, singer, etc. of exceptional celebrity, or one whose name is prominently advertised as a special attraction to the public...orig. Theatr.

["star, n.1". OED Online. December 2015. Oxford University Press. http://www.oed.com/view/Entry/189081 (accessed January 05, 2016)]

first appear in print in articles and publications about the theater, with reference to David Garrick:

(From "Memoirs of David Garrick, Esq.", an article in the July 1765 edition of The Gentleman's and London Magazine.)

The earliest quote given for the theatrical star sense (5a) in OED Online, dated 1779, also refers to David Garrick, yet (as indicated by the square brackets surrounding it) "is relevant to the development of a sense but not directly illustrative of it":

[1779 J. Warner in J. H. Jesse G. Selwyn & his Contemp. (1844) IV. 30 The little stars, who hid their diminished rays in his [Garrick's] presence, begin to abuse him.]

The next attestation I could find for 'star' in the theatrical sense is this from 1812:

(From Biographia Dramatica: Authors and actors. Longman, Hurst, Rees, Orme, and Brown, 1812.)

That use also predates the first attestation in OED Online purported to be "directly illustrative" of the theatrical sense (5a), a quote from 1824 that refers to a star of pugilism:

1824 Compl. Hist. Murder Mr. Weare 219 Carter..was at a loss for a star in the pugilistic hemisphere to produce him a crowded house.

I did not uncover any evidence that 'star' was applied to the divas of opera before its use for leading characters of the stage.

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For your (2), regarding the earliest application of the phrase "a star is born" to artists, that application seems to have begun or at least to have first become common with the release of the 1937 movie, A Star is Born. The script for the popular romantic drama starring Janet Gaynor and Fredric Marsh was written by William Wellman, Robert Carson, Dorothy Parker, and Alan Campbell. The plot concerns a young actress and an older Hollywood actor who helped launch her career.

Here is Dorothy:

(From "To Richard--With Love", by Dorothy Parker, reprinted from The Screen Guilds' Magazine, May 1936, in Celebrity Articles from the Screen Guild Magazine, Anna Kate Sterling, Scarecrow Press, 1987.)