So, for the first question, the answer depends on the situation. If it is a class you usually have at a certain time and you are asking another student in the class, it is okay to say “Will there be linguistic class tomorrow?” A better way is “Will there be class tomorrow?” However, if for example, you are new to a college or school, and you are looking for ANY linguistic class, you can ask anyone that attends the school “Hi, will there be a linguistic class tomorrow?” For the second question, both are **technically*** grammatically correct but the first is more concise (short and to the point), so saying “Are we having Mr Brown’s linguistic class tomorrow” is much better. However, the second statement can be used if Mr. Brown has never taught a linguistic class before but you heard he might be teaching one tomorrow. The main points of this very long answer are: 1) When you specify using “a”, you are not referring to a specific item. For example, “a linguistic class” can be any class versus “the linguistic class” is a specific class. 2) If you do not use “a”, then most likely the person you are talking to knows what you are referring to. 3) For your second sentence, short and precise sentences often sound better in English. This was LONG hahaha, so please feel free to ask me any questions if you have any doubts! I hope this helps.

I would likely say: "Do we have Linguistics tomorrow?" "Is there Linguistics tomorrow?" or (in my very sleep deprived state as a college student) "Linguistics? Tomorrow?" I'd probably not use the word "class" Similarly, I'd say: "Do we have class with Brown tomorrow?" or "Is Brown's class on tomorrow?" The whole phrase "Mr. Brown's Linguistics class" is correct, but in normal conversation I'd only likely say as many words as necessary to be understood. So "class" or "Linguistics" or "Brown's class" or "Brown's Linguistics" (if there were more than one Linguistics class).