"He likes to cook with a father."
Translation:Він любить готувати з його батьком.
I'm noticing that the indefinite article "a" is being used to translate what I believe would mean, "He likes to cook with his father" (or eat with her sister, or dance with his mother, etc.). Am I incorrect in assuming that this usage of the instrumental would be translated as I suggest? In Ukrainian one does not have to explicitly state his or her here as one would in English, am I right?
Absolutely. You don't have to use a possessive pronoun in those sentences. E.g. She's eating with her sister -- Вона їсть із сестрою, He's dancing with his mother -- Він танцює з мамою, He went on a trip with his family yesterday -- Він учора відправився у подорож із сім’єю.
The usage of the indefinite article in this sentence in English is kind of strange and unnatural, isn't it?