Does the iPad automatically stop charging when it is done

Yes. Actually, the charging slows down when it hits 80% and stops completely when fully charged. You can not hurt the battery.

Additional information below.

According to Apple:

Use Your iPad Regularly

For proper reporting of the battery’s state of charge, be sure to go through at least one charge cycle per month (charging the battery to 100% and then completely running it down).

Elsewhere, Apple elaborates and explains that two half-discharges (or four quarter-discharges, etc.) equals one full discharge.

http://support.apple.com/kb/HT4060

Yes. When a charge is complete, it will stop charging. Plugging it in over night is perfectly safe. I've charged my iOS devices and MBP that way for years.