Why wouldn't you use "với" but rather "cùng" here?
There is a subtle difference but it should also be accepted. Cùng means "together with" while với means "with". You can say "tôi đi cùng bạn", "tôi đi với bạn" or "tôi đi cùng với bạn".
You are right.
"với" = with
cùng = along with
If it rejected your answer, you should report.
In most cases, you can use "với" and "cùng" interchangeable.
Why not "Khi nào bạn nấu ăn cùng mẹ bạn?"
How would " When are you cooking with your mother" translate to in Vietnamese?
In this case, Vietnamese does not distinguish between 'when do you cook...' and 'when are you cooking...'. 'Khi nào' provides the future aspect.