Correct. In English, 'already' should be placed in this instance, at the end position (of the sentence) to express a degree of surprise. Hence, the translation should read: My father (has) fired the man already.
벌써 = already (as in earlier than expected)
이미 = already (as in "a done deal")
Generally speaking -되다 verbs are often passive forms of -하다. -되다 verbs are usually of the form "to get Xed" or "to be Xed" depending on whether it's present or past tense.
해고하다 - to fire (someone) 해고되다 - to get fired 해고됐다 - to be fired
I'm not sure on the form for "to get (someone else) fired".