Can I batch change more than one event type at once? I want them all to be the new, same type. Do I have to go into each event configuration separately to change the event type?
If the events are all part of a repeating event series, then you should be able to change them by changing one, and selecting the option to change all related events.
Otherwise, I'm afraid there's no built-in way to bulk change event types. If you know what you're doing with the API Explorer or MySQL, you can bulk change the event type on the back end - there's no real harm unless you have custom fields tied to the old event type, in which case they might show up unexpectedly on a report.
I've found the CSV import from the Fuzion team very helpful: https://github.com/eileenmcnaughton/nz.co.fuzion.csvimport/blob/master/README.
At least this allows you to import events... I've not actually used it to bulk change a field in existing events, but I 'think' it might be possible.
You would need to export the events in question, and in Drupal I use Views to do that. From that export you can make a csv with the correct field(s) and data, and on import create a field map to put the new data into the correct place.