Handle API query errors gracefully and inform the user of failures

We've had several instances recently of errors in API responses not being handled gracefully and not being reported to the user.
In several places, we have no error handling at all, so either quietly ignore errors or assume success, often resulting in unexpected behaviour.

Ideally, we would have a mechanism in the Admin Panel to allow users to report errors directly when they occur, along with the network request and response details, but a good start would be graceful handling of failures.