8-year-old kid

At least in American English, it's written without the "s."

In all varieties of English, American or otherwise, there are two ways of using two nouns to talk about time, distance and other forms of measurement:

Before the noun, using a hyphen to join the number and unit together: a ten-mile walk; a five-ton weight; a six-hour wait. In this form the singular form of the measurement noun (e.g. mile, ton, hour) is used. In the case of age, then a further hyphen joins the unit to the word 'old' - a fve-year-old child; a ten-month-old baby; a sixty-year-old book, a hundred-year-old egg.

After the noun, using no hyphens: he is six years old; the truck weighs ten tons; the rope is four metres long. In this form the plural form of the measurement (years, tons, metres) is used.

Thus "8-year-old kid" is correct in any variety of English. 'Kid' is an informal alternative to 'child' in most varieties of English, not just the US variety.

Some more hyphenated examples

a hundred-metre race
a 250-page book
a fifty-year project
a three-inch-high statuette
a five-metre-long boat

Thank you for your help.
Serge

Thank you for your detailed answer.
Serge