Questions on History of Advaita

[Sankar Jayanarayanan]

namaste.

I was just reading a book on the history of Buddhism and have some questions
regarding the history of advaita. Thanks in advance for your replies.

1) Were Shankara's original compositions "written" ? If so, are the original
   books preserved now? Or were his Bhashyas transmitted by word of mouth only?

2) I know that the Upanishads are said to be "unauthored", but was this opinion
   prevalent even during Buddha's time? If not, when did this "apaurusheya"
   business begin? (I have already read the previous thread on apaurusheya, but
   it seemed to focus on the belief around Shankara's time, not Buddha's time)
   I mean, it would have been mighty difficult to convince Buddhists that the
   Upanishads are unauthored!

3) Were there different branches of Vedanta like "advaita", "vishishhTaadvaita"
   etc. before or during Buddha's time? If so what were they?

Om namo nArAyaNAya

Kartik