Is it possible to have a PAdES signature with only a timestamp? Is it possible to have a PAdES signature which only contains a timestamp from a TSA? Does this prove the content of the PDF was unaltered since timestamping?

Or is the timestamp calculated on a digital signature and does it only provide valid proof for time?
Short answer: No. If you look at the PAdES specification, it is clear that each profile relies on at least one normal signature.

Longer answer: A timestamp and a signature serve different purposes. While a trusted timestamp does ensure the integrity of the data (alongside a notAfter date) it does not say anything about the authenticity of the the document itself, anyone can request a timestamp for any document.

The purpose of a normal signature (and PAdES is a signature standard) is to authenticate the author or issuer of a document.