I’m writing a science fiction type short story set in Germany (actually Stuttgart). To some extent the location isn’t that important- it could be set anywhere but setting it there appeals to me and gives the story a unique aspect. As someone who doesn’t know German and have only briefly visited that part of the country, how detrimental would that be when writing about the characters and their thoughts and dialogue? I guess I’ll be writing as an English native and not German - does that matter in the context of the story?
You are writing in English, your characters are going to speak English, but that is in fact a "translation convention", your characters "in reality" speaking German instead. Is that the issue you're asking about?
Such situations are in fact very common in writing. Consider a book translated from German to English - the situation would be just the same as you describe. And many books are written from the start just as you describe. For example, Jules Verne's Les Enfants du capitaine Grant is a book written in French about a Scottish family.
What would be jarring is if your characters start using turns of phrase that are very location-specific: 'Megabucks' would only be used in America, 'lass' is most commonly used in Scotland, etc. When you use such words, your story says 'Germany', but the language says something else.
I guess it is possible for a way of thinking, a way of responding to a situation, to be "very American", "not how a German would respond". Cultural differences do exist. You should do your research about this - talk to someone from Germany, read about Germany. This shouldn't stop you from writing, though. End of the day, people are people, and globalisation makes us more and more alike.