How does Sky Demon handle the updating of AIP plates (typically on change of AIRAC cycle) when they are amended or deleted or when new charts are added to the AIP? Is the update process automatic and coincident with e.g. chart updates, or do I have to do something manually?

We update plates every evening, but obviously the evening of an AIRAC switchover is the big one. In the morning of a new AIRAC day there will therefore be many plates on your local cache which will have been replaced. These will just seamlessly be re-downloaded next time you ask for them, but if you prefer to keep a cache of an entire AIP on your iPad, you'll want to do another Download AIP operation that morning to ensure you have a complete new cache.

So the ones on the iPad have to be updated manually? Had you considered an "update everything in the local cache" button? It could work selectively or simply download everything again.

That's essentially what the Download AIP function does. The forthcoming version also sports a Download All function which downloads all AIPs for the charts you have installed.

As an extra question to this, if I click off of the Tools->Download AIP dialog while it's downloading, does the download continue in the background? (on iOS)

No, that cancels the operation.