Learn Russian: У меня лишь один вопрос.

Yes, you absolutely can! Лишь means only or merely and is practically interchangeable with только. Лишь can sound slightly more formal or emphatic, but in everyday speech you will hear both used.

Why is the word есть missing from this sentence?

When using the у меня (I have) structure, Russian often drops the word есть if the focus is on how many or what kind of things you have, rather than the mere fact that they exist. Since the emphasis here is on having only one question, есть is omitted.

What case is один вопрос in, and why?

It is in the Nominative case (the basic dictionary form). Remember that the у меня... structure literally translates to 'By me [is]...' Therefore, the thing you have (один вопрос) is grammatically the subject of the sentence, not a direct object.

I am learning the negative pronoun ничто (nothing). Could I say У меня лишь ничто?

That sounds very unnatural. To say you have nothing, you must use the standard negative possession structure with a double negative: У меня нет ничего (using the Genitive form of ничто). The word лишь (only) is meant for specific quantities or things, like один вопрос.