Can it ever be grammatical to say “six feet five” instead of the normal “six foot five”?
Answer: Not really; if you say feet there you’d best add inches afterwards. Leaving it out doesn’t sound right to me anymore.
Does including inches at the end of “six foot/feet five inches” affect the grammatical number of foot to make it feet?
Answer: Yes, it sometimes can. See below.
John Clease is six foot five. (this is actually the weird one)
John Clease is six *feet five. (not grammatical for me but YYMV)
John Clease is six feet, five *inch. (not grammatical)
John Clease is six feet, five inches.
John Clease is six ?foot, five inches. (may not be grammatical for all native speakers)
Six-foot-five actor John Cleese towers over his cohorts.
At six feet five inches, actor John Cleese towers over his cohorts.
Towering over his cohorts, actor John Clease stands a full six foot five. (another weird one(?))
Not all these John Cleese examples follow the patterns we’ve seen before, and it isn’t clear why they differ. For example the predicate use can remain uninflected, while before all predicate examples inflected for number. That’s not true for babies: