Translation:Doctor Li is at the hospital.
wouldn't "in hospital" (and even "in the hospital" usually) imply that the person is there as a patient, and not just simply present there? whereas I'm assuming they mean the doctor is there working, not as a patient. in the US at least, saying someone's in the hospital generally implies they're there as a patient, whereas an employee, visitor, etc. is "at the hospital." I think that's why there's some confusion here. this isn't a hard rule of course but typically how those phrases are used contextually
正在 means "currently doing", like "我正在寫" (I am currently writing), you use 著 if you're doing something that is continuous, "我愛著你" (I am (continuously) loving you) if you want to say "I am currently at the hospital" you would have to put another 在 to show that you're there, ("我正在在醫院" (I am at/in the hospital right now)) otherwise if you say "I am hospital right now"
Perhaps not. However "in hospital " is grammatically correct in British English. But there is an important difference between being "at the hospital " and "in hospital ". The former would apply to a person working in or visiting the hospital. The latter would refer to an in patient.