Φτιάχνω=make and κάνω=do. There are cases where either of them works but in general φτιάχνω is a bit more creative. And for that reason possibly, in Greek you don't make a mistake, you just do it: "κάνω λάθος".

Without context either might be possible. Of course, the most usual use would be: "We're making dinner (as we do every day)." But also: "We're making the dinner (for the wedding reception.)" Let me see if I can fix it so it's "either / or". Well, that was easy. It was already done. So, that both "dinner" or "the dinner" would be accepted. I just added "the dinner" to the hints.

"We are making dinner" was accepted.

You can say "We are fixing dinner" but it doesn't mean "repair"; it's a way of saying "prepare": 5 : to get ready : prepare fix lunch https://www.merriam-webster.com/dictionary/fix

No. "φτιάχνω" literally means "make" and secondarily "fix" as in repair something. So you can make dinner, prepare dinner but never repair dinner.

"Δείπνο" is the main meal of the day while "βραδινό" is the evening meal specifically. Depending on where you live and circumstances the main meal could be in the afternoon or evening. For example we could speak of "Το γαμήλιο δείπνο" "The wedding dinner" which would depend on the time of the wedding.