We haven't tested this extensively, but in theory, yes! You can also set a higher batch size, which is the number of examples that are requested from the Prodigy server at once (default is 10).
The batch size is also used to determine when to request new examples from the server. If you manually hit save (or the app autosaves the current progress) and the connection fails, the annotations are kept in Prodigy's "outbox" on the client, and Prodigy will try again next time. So as long as you don't close your browser, no progress should be lost. This should also work if your Prodigy server crashes for some reason – as long as you keep the browser window open, you can always restart the server and save the progress.
Yes, ngrok actually works really well with Prodigy – see this thread and this cookbook section.
(Btw, just to clarify in case others come across this topic later: Prodigy itself of course works 100% offline, given you're on the same network. This question is mostly about to serving Prodigy on your own servers, and then connecting to it from a device with intermittent connectivity)