You can use those other forms and still be correct.

because you can then attach a subjunctive mood clause. "I was wondering if you would help me out with this", for example.

Hello Edward, all the forms of wondering you asked about are correct, but using "I wonder" as a way to begin a request is a very soft, gentle and polite way of asking for something. If you want to be more forceful or urgent when you ask for something, I would suggest using, "I need ..." since it expresses that more directly. Example: I need to know right away what time the event is. Hope that helps.

You can definitely use "I am wondering" or "I wonder"! I wonder if it will rain tomorrow. I'm wondering if the weather will be good. I'm wondering if elephants really do have a long memory. "I was wondering..." can also be used when you want to ask someone for something or a favor. "I was wondering if I could borrow your car?" Hope this helps! :)