We experimented with live sessions but weren't happy it was real-time enough to be more meaningful than after a short time delay, it's in our backlog to revisit when we can spend time on performance though!
At the moment it's not set to be triggered for specific sessions, none of our early users have requested that but it would be pretty easy for us to implement :) At the moment we're letting our most engaged users take a very active role in determining what comes next e.g. we rolled out IP screening (all happens client-side) and email/domain name screening yesterday as per a request we received yesterday morning. If you sign up we could get triggered recordings in for you super fast ;)
No open API access, we have discussed it though. Recording link sharing with private, or public links, and mp4 and gif (snippets) export is next up.
Sounds like you're a FullStory user at the moment, which features do you currently get the most value from?