I would take a look at how Facebook implements its notifications as many users will not only have experience with it, but Facebook has also spent many worker hours developing and refining it to what it is today.
Q: Are there preferred locations to display the notification?
Since you mention the user can go on and do other things, the notification should be out of the way, unless it needs their immediate attention. A good spot would be in a top bar (where most interfaces have notifications).
Q: Should they expire automatically? (timeout vs wait for acknowledgment)
This is heavily dependent on use case and I would suggest requires user testing to determine your typical user's behavior in response to seeing a notification.
Q: How to handle notifications after a user has left the input screen/process?
Since the user can go on and do other things -- this includes leaving their device. It would be cumbersome for the user to have to re-enter and/or re-submit the information their verifying. I would suggest that they can view a history of all of their notifications (like Facebook's click to view the most recent ones and a "See All" to view the full list).
Q: How to handle event compounding? (eg process 1 completed, process 2 completed) Do they stack? Does #2 replace #1?
Again this is heavily dependent on user behavior and their workflow with this app. Since they have the ability to do other things, this may involve submitting another input. They might even submit several inputs back to back to be checked -- in which case they would not want to be interrupted. If this is the case, they will most definitely need to be stacked.
I see several options for you here, but I would try a notification cue (like Facebook's desktop site -- when you click the icon, it shows you the most recent notifications first with an option to view the full list). The most recent ones could be tied to the current app session. Again, user testing is important here.
In contrast, StackExchange's notification system shows a long list of notifications (might be all of them, not sure as I'm new to the platform and don't have a long history).
Both show a change in color and a number over an icon.