1. we became friends the other day = perfect sentence 2. 'the other night' = means a couple of nights ago. NOT last night, the night before that. It's a little confusing. Just remember; if it didn't happen last night, it happened 'the other night'

It refers to a recent past, just some nights ago. I think what's important is to understand that "the other day or night" is to refer to a recent past instead of a distant one. I hope it helps! :)

The sentence is fine and the correct tense is used. I would disagree with above comment because `other night` would typically refer to more than two nights ago. If you wanted to talk about two nights ago you would normally say couple nights ago.