Can something be private and owned by a state? Isn't the definition of private: non-public? And wouldn't the government in a democratic state count as public?
"Public enterprises" are different kinds of enterprises, with different structure and governed by different regulations/laws than private enterprises. "State owned private enterprises" have all the structure etc of a private enterprise, their only difference being that they are owned (fully or by majority) by some state body.
"Public company" is a very specific thing; the BVG is state owned, but very much not a public company.
Yes, it can. Many public services in Germany are legally private companies, which just happen to be owned by public entities. Main reason seems to be that they can act more flexible and independent of public rules. On a more nasty side, it's also easier to sell them partly or full, if necessary. And in some cases they simply started as private companies and were bought up by the state for various reasons.