Yes it's a general rule. We don't use the article in front of the profession with the verb "ser", but we use it if we want to specify.
El es el profesor de matematicas=He is the Mathematics teacher (You are talking about a group of teachers and you want to specify that person is the mathematics teacher)
El es profesor de matemáticas= He is a mathematics teacher.
And with the article "un" the only reason that I can think now is for emphasis
The question marks :) Much like in English, where you can ask a question as "Your dad is a teacher?" and "Is your dad a teacher?", in Spanish you can ask it as "¿Es tu padre profesor?" or as "¿Tu padre es profesor?" (which has the same word order as the non-question statement).
Yes, but I think proper inflection is more important in Spanish than English (but that could just be my foreigner bias). When I lived in Spain, I remember plenty of occasions when I was speculating out loud, but my compañeros thought I was stating fact instead of asking a question. I often had to repeat myself with exaggerated inflection.
This might not make much sense to anyone else, but my method of remembering this was to imagine an equation of sorts between the two translations with the "r" and the accent mark. If there's an accent mark, then you drop the r (so tú = you; in a way, the "r" comes over and becomes the accent mark) and if there isn't, you include the r (tu = your; the accent mark jumps over and becomes an "r"). There's only ever one at a time: either an "r" or an accent mark. That might sound confusing but it worked for me.
So the most accurate answer 'Is your father a professor?' is being marked incorrect. Thanks DL.
Not quite understanding why 'the' couldn't be used here as well. If my friend and I walked into a lecture hall and I see my friend's father at the podium would I be wrong if I said tu padre es profesor while looking at the podium meaning 'your father is the professor ' giving the lecture today?
Yes you are absolutely right. I should redeem myself by giving you this link that further explains