It can mean both A broadcasting company and THE broadcasting company. Korean can be more vague than English and your translations are assuming meanings that aren't really there and it's annoying.
"I work for the broadcasting station. " is not correct?
에서 is marking 방송국 so 방송국 is only being referred to as a place and not an entity.
Ya it is
"I work in broadcasting" is apparently wrong
Drop the 국 and maybe you're fine. It sounds instrumental to me, so I'd have thought 방송으로, but yours is more common and favored by Google Translate.
For this one I thought "I work for a broadcaster." Don't try it though . . .