So how is your 10th grader doing in school here? S/he should be at least a junior when he gets back to school this fall. If the high school has had German exchange students, they should know what to do. If he was near the top of his class here, I would consider placing her/him as a senior. This is based on my experience (albeit in the 80s) when I spent a year in an American high school at 16 and after finishing 10th grade Gymnasium. I was placed as a junior but allowed to take advanced placement senior classes. I was in general junior English for a few weeks which was boring me to death so allowed to switch to senior AP English. I could have graduated after that year if I had chosen to take one semester of American history and of government, but I chose fun stuff like debate which we didn't have in Germany or theater.
They didn't ask for curriculae but maybe this was because the school was in a college town and they'd had plenty of international and exchange students. I think they want to know what topics were covered per subject. I wouldn't bother asking the schools but rather look here:
and maybe use deepl to translate (also, feel free to PM me to look over the translation - as a native German speaker and near native English speaker who knows both systems I can probably spot errors). They would see e.g. that 10th grade means advanced algebra and trigonometry were covered and your kid is ready for AP calculus.
edit: I should ask, is this a G8 (12 year) or G9 (13 year) gymnasium? In G8 (still the norm in BaWü), 10th grade math includes some calculus:
I also noticed your other kids are a tad on the old side in the German system and will certainly be in the US. I would definitely place them in the same year they would be attending in Germany this fall. I don't think there will be issues. Having completed 8th grade Realschule, they will have covered Algebra I and Geometry I which was enough to graduate high school in Indiana when I was there With the German system emphasizing essay type written exams and presentations for oral grades, they will ace English and Social Sciences, I am sure of that. And they can potentially get credit for German. Being bilingual or multilingual with proper command of grammar in at least two languages helps cognitive skills and will go a long way in any school.