Is there a way to determine the last time replication (push or pull) occurred? By default, is there a last modified timestamp on each document? If so, is there a way to access this from the sync gateway Admin REST API? Or would I need to manage that (by created a last modified field in my documents)?
No. You’d have to remember this on your own.
By default, is there a last modified timestamp on each document?
No. You can add it yourself. Just remember that the time a doc is created/modified is not necessarily close to the same time it arrives at the server, if the client was offline. Also, you can’t trust the clocks on clients; they can be accidentally or maliciously inaccurate.