"This classroom is very big."
クラス is techincally "class" and not "classroom" (きょうしつ). If you used きょうしつ then you couls be implying that there are a lot of people in your class, which is not what you are trying to say.
From what I understood in the lesson with せまい and ひろい , they are preferred over 大きい and ちさい when talking about the size of a certain space. Is that true and if yes, are the latter two just uncommon or flat out wrong to use in this situation?
I would use せまい or ひろい in this sentence. These are better for describing a space that you are in.
While inside a space:
This room is very narrow/cramped. このへやはとてもせまいです。
This room is very big/spacious. この部屋はとても広いです。
大きい and 小さい are better used for something that you are looking at from the outside.
While you are not inside a space:
That house is very big. そのいえはとても大きいです。
This apple is very small. このりんごはとても小さいです。
In a direct translation 大きい and 小さい are technically correct, but in this instance it doesn't sound natural.
クラス = class (the course, or the collective body of students and the teacher)
きょうしつ = classroom (being the physical room in which the class takes place)