desu is the verb (it literally translates to "it is"), so including it is technically more correct. However, according to my irl Japanese teacher, casual spoken Japanese often leaves out the verb, especially if it's desu. So just plain mochiron is probably more common usage, even if it's not grammatically correct.
I've just started learning, so I can't answer with any expertise, but I'll throw my two cents in, since this question has yet to be answered:
First, I'm assuming you mean "もちろん", as that is what is in the lesson/at the top of this page, not "もしろん".
In a reply to another comment, moderator Swisidniak explains that "もちろん" (or, at least the kanji form, "勿論") literally means "no argument".
The "か" at the end of the statement "もちろんです" transforms it into a question.
My best guess is that "もちろんですか？" means "Is there no argument?", or roughly, "do you agree?" or "got it?".