"A classmate and I was" vs "A classmate and I were"

And links two things (e.g. a classmate and I), and results in a plural subject (A classmate = 1, I = 1, so a classmate and I = 1 + 1 = 2), so were is correct:

Awarded such and such scholarship in 2011. A classmate and I were the first students from my college to be awarded this scholarship.

The rule you refer to applies only to figuring what case of pronoun to use (e.g. I or me), not to whether you should use singular or plural (e.g. was or were). When you use the rule to figure out which of I or me is correct, you should alter the number of the verb (were becomes was) and direct object (students becomes student) when you take out the other person:

A classmate and I were the first students becomes I was the first student (correct).

A classmate and me were the first students becomes Me was the first student (incorrect).

When you add the classmate back into the sentence, you can be assured of the proper pronoun (I, not me).

the rule I've been taught is to take the other person out and use the words that make sense about just you.

This rule should be applied to what pronoun you're going to use, not what verb form: should you say "Bob went to the store with Mary and I"? Take Mary out of the sentence, and you're left with "Bob went to the store with I" which is obviously wrong.

For pluralization, if you can reasonably substitute a plural pronoun, then you should use a plural verb. "A classmate and I" -> "we" -> "we were" -> "A classmate and I were".