Regarding (2), some evidence that Baldwin refers to some sort of $L_{\omega_1 \omega}$ version of Morley's theorem, rather than just an alternate proof making use of $L_{\omega_1 \omega}$ machinery, comes from a [1970 survey by Keisler himself][1]. He mentions that "various forms" of Morley's theorem were extended to $L_{\omega_1 \omega}$ by "Choodnovsky [sic], Keisler, and Shelah, 1969" (p.149) though no citation is included in the references. And a look through the [Shelah archive][2] seems to turn up no relevant joint work with either of the other two.

I don't have a copy on hand, but one promising source for clarification (beyond inquiring with Baldwin about the content of his slides) is Keisler's 1971 book *Model Theory for Infinitary Logic*, which likely covers the result(s) in question such as they are; and though perhaps only a coincidence, that does match the year Baldwin's slides assign to the matter.

**ETA:** Baldwin's [*Categoricity* book][3] confirms both the nature of the result and his direct source: "Keisler [Kei71] generalized Morley’s categoricity theorem to sentences in $L_{\omega_1 \omega}$, assuming that the categoricity model was $\aleph_1$-homogeneous" (p.22). Though Baldwin points to Keisler's book as the basis for transferring Morely's theorem to infintary logic, he also attributes most of the machinery to Shelah (p.xi).


  [1]: http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0141.0150.ocr.pdf
  [2]: http://shelah.logic.at/
  [3]: http://homepages.math.uic.edu/~jbaldwin/pub/AEClec.pdf