"She asks him to continue."
Translation:Hon ber honom att fortsätta.
Does a preference exist in Swedish communication, between using "Hon ber honom att fortsätta." or "Hon ber honom fortsätta."
Or are they both common?
It depends on the verb. For some, both are possible, for some you can’t have att, for some you have to have it. In this case, you could say either.
Why is frågar in this case not accepted as a translation of asks? Is it because she doesn't ask for information, but for him to do something?
Yes, that is exactly the reason. When you ask someone for information, you use "fråga". But when asking someone for an action, you use "be".