Can a PDF file contain a virus?

There are many features in the PDF that can be used in malicious ways without exploiting a vulnerability. One example is given by Didier Stevens here. Basically he embeds an executable and has it launch when opening the file. I am not sure how today's versions of readers handle this but its a good method of using PDF features in malicious ways.

Yes it can. PDF is a rich format that aside from static content, can contain dynamic elements. The latter can for example contain JavaScript, and other elements. Modern PDF viewers tend to warn the user about potential malicious activity though.

If you want an example malware, check out pidief.

And generally PDF malware will predominantly be just the dropper, not the payload itself.

To learn more on the vulnerabilities associated with pdf files and ways of detecting them before they do any damage read this kali documentation on peepdf.

Yes it can.

Whether a file is malicious or not, does not depend on the file extension (in this case PDF). It depends on the vulnerabilities in the software which will be parsing it. So for example, PDF reader that you are using potentially contains a buffer overflow vulnerability, then an attacker can construct a special PDF file to exploit that vulnerability.

Consequently, to guard against such attacks is also easy, just ensure your PDF reader is up-to-date.

A simple google search landed me up on the SANS Institute's overview of PDF malware, which seems to be good to start with.