Is there a reason to clean the sensor at startup, shutdown, (or both) automatically?

Apparently auto sensor cleaning is effective at reducing sensor dust during normal usage, so it makes sense to use it unless you like dust.

Startup and shutdown are just good times to do it, the camera is on but not in use and it's just before you take pictures (reduce dust before it becomes visible) or after you use it (after you let some dust in).

The cleanup takes a little bit of time, so if you want the camera to turn on as quickly as possible you don't want auto cleanup on startup - but apart from that it does seem like an exceptionally good time to prevent dust from appearing in the photos.

Can the sensor be damaged in any form from repeated cleaning like this?

Considering the amount of cameras sold with auto sensor cleanup enabled by default if it caused sensor damage we would have known by now.

If you do not have it ran automatically, how often should you clean the sensor using the automatic method?

Like any cleanup it depends on the camera sealing and your working environment.