questionというのは可算名詞だからです。 可算名詞とは数えられる名詞で、一つの質問ならa question複数の質問ならquestionsとなります。よって、 a をつけない場合は、questionsとすれば○です！
It seems that when saying "something has something," you need 持つ to be in the present continuous form (In English, it's like "am eating," "am walking," etc. It describes an action currently being executed), which is 持っています or 持っている (depending on how formal you want to be). I know "am having" sounds weird in English, but it sorta makes sense in Japanese; you currently have blank.
持つ is used for possession, ある is used for attribution. You could use 持つ in a case where you were carrying a something with a question on it to say that you literally had that question in your possession, but not in a case where you simply were asking a question.