Can I set an upload to expire after a certain time period?

Click on the pencil on the image thumbnail and choose for it when to expire.
Available options: 5 minutes, 30 minutes, 1 hour, 2 hours, 6 hours, 12 hours, 1 days, 2 days.

Can I set a password for my uploads?

Yes, but you need to register an account.
NOTE: Email is not checked for validity, and there is no confirmation of any kind.

What steps should I take when uploading private documents?

Make sure it's set to auto delete, or as a private upload. Public images can be accessed by "guessing" the public image ID. Uploading it privately means the image can only be accessed by direct link or as an album with a password, if you choose to share it.

Support: anonimages@protonmail.com