The man's name Осип is a form of Иосиф. Is it a nickname, a diminutive, a regionalism, or a register change? Would someone named Иосиф potentially also answer to Осип, as with Joseph and Joe?
It's neither a nickname, nor a dinimutive. It's just a form which became distinctive from the (borrowed through Greek) Иосиф and happily co-exist with it just like Johannes co-exist with John.
While it indeed initially was introduced at lower-class usage I'm not sure it can be treated as regionalism - I'd rather say it is Russian vulgar version of the name. Just like say Йозеф.
As of the second part of your question - yes, in colloquial speech Иосиф could be addressed as
Осип if he didn't mind - nowadays it's pretty obsolete.
There's a famous Russian-speaking poet of Jewish origin, Осип Мандельштам, who actually was Иосиф. But this name can be use as a separate one as well.
Actually, there are some other examples when colloquial form of name became a separate one. For instance,
Егор initially was a form of
Иосиф is not really a Russian name. It is a Jewish name, which can be encountered among Russian Jews moderately often and never among ethnic Russians.
Осип is a Russian name, but it is very archaic, outdated and not widely used.
If I encountered a person named Осип I would suspect a family of Russian Orthodox conservatives, priests, Old Believers, peasants, whatever of this kind.