When a specified noun is preceded by a noun or a pronoun, then 的 can be omitted
1. these two dogs of mine. 我这两条狗
In cases where two nouns have a close "personal relationship, or when the first noun or pronoun belongs to a group indicated by the second noun, 的 is often omitted.
1. Her mother is an American. 她妈妈是美国人
2. His colleague is on a business trip. 他同事出差了.
As for your question, 1） 我（的）美国朋友, 我同学（的）家 both are omittable. 2）
2) 同学=同班？ Logically speaking, 同学 > 同班 in the scope: you can learn together without in the same classroom.
As a result, 我同学家 has been used about 228,000,000 times on the google search, while 我同班(同学)家 7,260,000。
And it should be noted that when you mention 同班, you mean that you are 同班同学(very specific), learning together in the same classroom.