If I get several links from authority sites that aren't really related to my site's topic, would they count towards an SEO benefit? Or worse, could it have a negative affect on my rankings?
Links from authority sites will indeed give you juice for SEO. But, links from relevant authority sites will count far more to your backlink profile.
I think a better way to go about link building is to think about direct audience. Just ask yourself: Does this website have an audience that could find value in my product?
If the answer is yes, go for it! Don't worry too much about whether it will help with Google rankings. It's better to get links that give a direct, immediate benefit to your sales. Google rankings follow suit automatically.