Greetings
The majority of student deals are open to ALL students, regardless of residency or citizenship. The more common restriction is that student/concession rates are limited to people under age 26. Or that all citizens, regardless of age, get a discount. The only time I've ever run into citizenship/residency limitations is for transport and that's because the regional/national government is subsidizing the cheaper fares.
The best form of student ID is an ISIC Card, because it's recognised just about everywhere, so there's no confusion.
I think your 10 year old should be fun unless he/she looks quite old, and then proof of age might be good for under-12 or similar discounts. The 15 year old might want to get an ISIC card if he/she looks old because not all 16+ year olds are in school in Europe.
Kate