Extract framework out of app

There are a few bits and bobs we can extract into a reusable framework. Things like the:

- ~Common Log Formatter~ (already abstracted into a middleware in #31).
- Router
- Content Negotiation (JSON, ICS for calendar stuff)
- Error monitoring and issue opening in GitHub

Review the app structure and the remaining Router, content-negotiation, and error-monitoring items listed in the issue; the common log formatter is already covered by #31. Define the reusable framework boundaries before changing code, with the remaining capabilities extracted from the app and usable independently as the completion criteria.

Written by the indexing model from the issue text.