Mind the placement of ‹s› after ‹cook›.
“my cooks cook pho for me.” ≠ “my cook cooks pho for me”
Easy to make an accident and lose a point…
technically your first sentence is not wrong. you don't always need to explicitly express the plural to make it plural. Duo does want us to use các or những though.
no. said that way, I wouldn't understand it as he is cooking for you [the speaker]. instead it would probably mean he is cooking for himself to eat, as mình is mostly taught as a reflective pronoun in Duo (the reflected subject is "đầu bếp" not "tôi", as "tôi" along with "của" is only a subject complement modifying "đầu bếp" and can be left out without changing the essence of the sentence).
however, mình could also refer to the "we" that includes the speaker and the listener. I tend to think of mình this way. "my cook cooks pho for us." (plus, as he is my cook and I probably hire him for this job, it is safe to presume him cooking for us rather than for himself)
in any ways, "mình" does not refer to the speaker in a sentence formulated like that.