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".
Wouldn't "We're making dinner." be a more appropriate translation? One wouldn't really ever the "the dinner" like this without something more following.
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.
I have always used/heard "φτιάχνω" as "I fix" (such as dinner or something that is broken). "We are fixing dinner" was marked wrong - should I not use it this way?
No. "φτιάχνω" literally means "make" and secondarily "fix" as in repair something. So you can make dinner, prepare dinner but never repair dinner.
Idiomatically "fix dinner" is used. It is American usage and does not refer to "repair" in this case.
"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.