When I flush the toilet, after the tank fills, there is a loud shut off knocking sound. This happens in both bathrooms. Also, both toilets ghost flush during the night several times--first in my bathroom and then a minute later in the main bathroom. What can I do?

The noise you describe is water hammer due to the abrupt shut of the ballcock. It can be eliminated by adding a screw on anti hammer device like those sold by Sioux. What is a ghost flush?

Sioux Chief Plumbing Parts & Accessories

I can't figure out anything that would flush a toilet automatically during the night, but I's like to ask a question:

1) Is this a pressure assist toilet2 Plastic tank inside tank.

And I have a suggestion in addition to bb's.

What's the house water pressure?