Going by public documents, it seems like the intention was to have a separate organization pretty early on (although I can't say about the very beginning since I wasn't involved).
The Articles of Incorporation for CFAR (called Feynman Foundation at the time) are dated July 18, 2011, and 501(c)(3) status was approved on July 26, 2011.
MIRI's 2011 strategic plan (updated August 2011; unclear when it was first written) says "Encourage a new organization to begin rationality instruction similar to what Singularity Institute did in 2011 with Rationality Minicamp and Rationality Boot Camp."
The Minicamp took place in May–June 2011, and it seems like the Boot Camp took place in June–August 2011, so it looks like by the time the first workshop finished and the second was in progress, the plan was already to start the new organization. However, it's still possible that there was no plan for a separate organization before or during the first workshop.
MIRI's December 2011 progress report also talks about plans to create the "Rationality Org".
I also wrote a timeline of CFAR a while back, which has more links.
Yes. Or at least, becoming a distinct organization was already the plan when I got there in early 2012: get a group of people together to create a rationality organization, initially rely on MIRI for institutional support, become an independent organization some months later once all the pieces are in place to do so.
When what would later become CFAR first started as a series of rationality workshops run under the purview of MIRI in 2012, was it always the intention of CFAR's founders to have it be a project/organization distinct from MIRI since its beginning? Or, did CFAR's founders decide to incorporate CFAR as an organization separate from MIRI after it became clear the rationality workshops weren't a fit project for MIRI itself later on?