"Children under ten years old."
In this sentence 'alt' isn't even necessary, since it's more like: Children under the age of ten. If you wanted to have 'alt' in the german sentence, it would be: Kinder, die unter zehn Jahre alt sind. 'alt' always needs a form of 'sein' to work properly.
Apart from that, it's also pretty common and I thought you would do that in English, too: How old are you? – I'm ten. Ich kann dir kein Bier verkaufen, du bist noch nicht 16. – I can't sell you beer, you're not yet 16. Would you actually add 'years old' in English?
Yeah, it depends on the situation. Under more formal writing or speaking one might be inclined to say the full sentence, "he's 25 years old", but quite often people respond, like you said, 'he's 25" or "he's not 21 yet" (I wish here was as cool as over there :( ).
But do note that when using age as an adjective, people almost always say "he's a 25 year old male", "she's a 32 year old accountant who lives in Germany", etc.