I think the breaking up line implies that a) the relationship between an hero and their sidekick(s) is a stronger bond than simply being coworkers, since they trust each other with their lives regularly and b) the reason why AM and his sidekick aren't working together anymore is deeply personal, and they haven't parted on good terms despite still caring about each other.

About Bakugou and his way of speaking... I sort of feel like in Japanese his rude way of speaking is supposed to sound as out-of-place and tryhard-ish as his constant F-bombing sounds to us. He's a teenager, after all.