国家 vs. 国, when can 国 be used alone?
You probably mean 国家 instead of 家国. 家国 doesn't exist.

你是哪国家人吗  is not correct (nor is 你是哪国家人), but you can say 你是哪国家的人 or 你是哪国家的 (meaning: from which country are you).

In 你是哪国人, 国 is not really  on its own. You can treat 哪国人 as a unit (which is a short form for 哪国家的人).

In oral Chinese 国 is seldomly used on its own (but every character however has a meaning, and 国 on its own also means country), but is used in for example 国内 (inside this country, typically used to mean within China, 内 means inside) or 国旗 (national flag, 旗 means flag). The list of such examples is long.