there is definitely an availability API. I helped lobby for it! We also implemented it at Hipcamp
(If you can't find a signup form on the rec.gov website just contact them directly and ask for access. I think their team was rolling access out slowly as they hadn't finished building a fully fledged login system but last I checked they do want 3rd parties being able to build on top of that data)
Not yet. It's part of the RFP spec that was written by USDS in the contracting phase. Unfortunately the language is non-binding so it wasn't introduced at Go-live for the new rec.gov site. It's in good faith that a booking piece will come, and much of the details need to be negotiated between govt, Booze, and 3rd parties since it is uncommon (unprecedented?) to have startups accepting payments on behalf of the federal govt (and in effect, on behalf of the treasury). Implementation details, revenue and commission details, etc are all to be worked out; the most important thing is that all parties get a fair shake (i.e. terms for one 3rd party extend to all 3rd parties so it's an open system instead of a closed partnerships-driven system). So far I've been impressed by the open communication of the Booze team and all signs point to their being good faith actors.
Note: If anyone has examples of startups accepting payments on behalf of the government please include them here. It helps with research. (Tax filing companies are the ones that come to mind... looking for other examples.)
Edit: the National Park Service also has its own API: https://www.nps.gov/subjects/developer/api-documentation.htm