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?".
ござる (abbr. of ござある) is a verb "to be, to exist" which itself is considered archaic. It is an honorific form of verbs いる・ある"exist". The polite form conjugation ございます is still often used in polite set expressions and the construction でございます is used in very formal/polite speech/literature and is a more polite form of です. You would use it in situations where you want to be extra polite, more so than just ordinary daily polite conversation. For example you may use it with your boss, or anyone in the customer service industry would use it to be extra respectful to their clients/customers.