What's the best option available for chat embedding right now? E.g. if I want to build Facebook's chat feature or Fiverr's Inbox feature and have it integrated into my main web app.
We haven't published an official embedded chat SDK for Matrix yet, but there are a bunch of options from the community. https://gitlab.com/sanlox/tangent is one, which is very new and immature but looks to be headed in the right direction.
Alternatively, you could take a scalpel to matrix-react-sdk and strip it down to do what's needed.
Maintaining UI SDKs like this is very time consuming, and we're currently putting our time into ensuring matrix-react-sdk (and thus Riot) pull their weight against Slack & Discord etc.