For this particular verb ("to own") the progressive "is -ing" is not used.
For verbs that describe actions (like "to run"), English tends to use the progressive to describe what is happening right now, and the simple present to describe habits or repeated actions. For example: - He is running (right now) - He runs (every day)
For verbs that describe emotions or states of being (like "to want", "to own", "to live (in a place)") that don't have clear beginnings or endings, typically only the simple present is used. For example: - He wants to eat dinner (not "he is wanting to eat dinner") - He owns a dog (not "he is owning a dog") - He lives in New York City
That last example is a bit tricky -- it would be correct to say, "he is living in New York City," but would strongly imply that this is a temporary thing ("he is living in NYC until/while something happens"), or possibly that he has recently moved (and we are contrasting where he lives now to where he lived last time you asked).
You have one letter wrong in the word "allowed". In this context, I allow it :P
Not direct synonyms, no. They express a similar idea, but from opposite perspectives, so you'd have to switch the roles/positions of the two things as well if you used the other verb.
Kokken eier en restaurant. = The chef owns a restaurant.
En restaurant tilhører kokken. = A restaurant belongs to (is owned by) the chef.
As far as whether that second sentence is natural-sounding, I have no idea. That's the grammatical explanation though.