Learn Russian: Директор недоплатил рабочим за прошлый месяц. - The director underpaid the workers for the last month.

No, here недо- is a single, continuous prefix indicating that an action was performed insufficiently. So недоплатить means 'to underpay.' If it were written as two words (не доплатил), it would mean he 'didn't pay the rest' or 'didn't pay the remaining balance.' The prefix недо- focuses on the action falling short of the required norm.

Why is рабочим in the dative case?

The base verb платить (to pay) and its prefixed forms like недоплатить take the dative case for the recipient of the payment. You are directing the payment to someone.

Рабочим has an adjective ending. Why isn't it a standard noun?

Рабочий (worker) is a substantivized adjective. This means it functions completely as a noun in a sentence, but it still declines using adjective endings. Here, it takes the standard dative plural adjective ending -им.

Why do we use за прошлый месяц instead of в прошлом месяце?

If you used в прошлом месяце (prepositional case), it would mean the act of paying physically took place last month. By using за + accusative case, you indicate exchange or compensation: the payment is for the work they completed during that month.