Some professors and university personal always address their students with "du". Should I as a student reply with "du"? If not, how can I ask them if its okay?
In my experience the "asymetric Du" (as I call it) is somewhat of an old concept and is going out of style lately.
Normally the Du/Sie between adults is symmetric, so if your professor says "Du", then so can you.
But traditionally in a hierarchical settings the "higher" person sometimes had the benefit of saying "Du", while the "lower" person was still expected to use "Sie".
From your description it's hard to judge what is expected of you here.
When I was an apprentice in Swabia it this asymetric Duzen was perfectly normal. The foreman would alway say du and we would siez. That changed when I was finished. You gained the priviledge to be symetrically adressed.
Using "Sie" is appropriate if you have a formal relationship to the other person, if the other person is older, in a higher position (e.g. your boss). It means you respect the other person. If a person tells you only the last name you should use "Sie". (I am a native german speaker and it's much harder for me to insult a person I address formally)
Using "Du" is appropriate if the other person is much younger (you would never say "sie" to a child), if you are friends, or if you are related. You would also use "du" if you want a personal relationship, so if you are speaking with other students or girls (in your privat time, not in office) you are interested in, you should never use "sie". If a person only tells you the first name, it is the same as offering "du". (Formerly children were used to adress their parents formally, but that's a long time ago.)
If I were in your situation, I would always use "sie" if I spoke to professors. If it's okay for them if you use "du", they will tell you. As you are not a native speaker it would also be ok if you ask him, but I wouldn't do so with my professors. I'd wait until they offer me to use "du".
Asymmetrical "du" is quite normal if the other person is much older or definitely higher in hirarchy.
"Knigge" is a book which answers many questions to good behaviour with other people. According to them, the person who is higher in hierarchy offers the "du" (source). If you already used "du" and the professor didn't tell you to use "Sie", you should continue to use "du". If a person askes you to use "Sie" or "du", you should definitely do so.
Some of my personal experiences: When I was in school in Bavaria, every teacher used "du" and every pupil "sie". Some of my former teachers offered the "du" after I got my final exam, but it were always the younger teachers. Every teacher asked us if it is still okay if they continue using "du" after we attained full age.
While I was doing community service (every German has to do either community service or military service) I used "sie" for all of my co-workers, as I was 20 and all of them were at least 30 and they had much more experience in the work than I had. All of them offered me to use "du" on my first day, except the boss. She was very young, but her position was higher. I guess she didn't offer me "du" as she had to tell me if I did something wrong and it's easier to do this on a formal basis.
I recently experienced that members of different political parties are used to a different behaviour. If you are on a meeting of the SPD it is very likely that they use "du". If you are on a meeting of the CDU it is very likely that they use "Sie".
A friend of mine who has a very good feeling for beeing politely uses "Sie" when she speaks with other students she works with. I was quite surprised as I would never use "Sie" if I spoke to other students in roughly the same age.
It might depend on the region. Here in Vienna, "du" is always mutual at university, so, yes, you would use "du" on a professor that adresses you with "du". There are some professors who prefer mutual "du" and some who prefer mutual "Sie", but I haven't encountered asymmetric "du" so far (at university), and it would be perceived as odd or impolite by many students.
If you are unsure about the local customs, I would recommend to ask a fellow student that has been around longer than you have, or some other third person, for example, the professor's secretary.