I'm fairly new to ballroom and don't know all the rules yet. Can one wear a strapless gown for a smooth competition or is that too promlike? Thanks in advance

One of the things that announces to everyone that you are a "newbie" is to wear a strapless dress. Invariably, the woman spends her time tugging it up, and she typically limits her movement in order to keep her dress up.
A lot of people discount the following, but my suggestion is to wear something that has the right kind of "weighted movement" - a leotard and a character skirt would work. And I DON'T mean some tired, worn-out set of practice wear. You can get some fresh and interesting selections at ballet shops or at online dancewear sites. There are some sites that specialize in ballroom/latin practice/performance wear, if you have a slightly larger budget.

I have been to many comps and I have never seen anyone wear a strapless dress. If the dress doesn't stay put, you can be pulled off the floor and told to change. I have a friend who was wearing a dress with one over -the -shoulder strap. The other side slipped down a little too low and the scrutineer told her to change. Look at ballroom dresses on ebay. There are a lot of choices and some very good prices. There is also the possibility of buying a used dress. As Clary said, you don't want to wear anything that screams 'newbie'.

You need to ASK what your instructor wants you in. If your dancing American Smooth some instructors don't want to see bare arms. I know some that require you to wear opera length gloves, but they had better not bag or bunch. If you have gontlets with a finger loop made, again they had better fit tight enough around your upper arm so as not to slip down.