"This is my friend in the class next door."
Yes, となりのクラスの友だち literally means "friend of class of neighbour." The の particle can be used as a possessive (あなたの本, "your book") which is usually the first thing people learn about it, but it's also used to link nouns together, as is the case here; rather than "my friend owned by the class" クラスの友だち means "my friend from the class."
となり is slightly different in that it's often classified as a "no-adjective;" if you use dictionaries like jisho.org you'll probably have heard of those. It works pretty much the same way, though: となり means "neighbour," となりの means "next door."
The word you were looking for is probebly 授業 (class)? 教室 means classroom.
授業 and クラス are very simular (tbh i keep confusing them to)
Quote: 授業 is a lecture in school where teachers teach something to their pupils. クラス is a unit of group of pupils organized to take 授業.
I also wondered about the use of "this" in the sentence and whether it would be used in Japanese. Apparently こちら is what they use. This link explains it some: https://japanese.stackexchange.com/questions/6845/a-polite-way-to-say-this-person