Learn Romanian: Băiatul a căzut pe ambele coate.

Yes! This is a great example of a noun with multiple plurals depending on its meaning. When cot refers to the body part (elbow), the plural is coate. When it refers to a bend or turn (like a bend in a river or a pipe), the plural is coturi.

Why do we use ambele instead of ambii here?

Cot is a neuter noun (un cot, două coate). Remember that neuter nouns behave like masculine nouns in the singular, but take feminine agreement in the plural. Therefore, it uses the feminine/neuter plural form of "both", which is ambele. If it were a masculine plural noun, you would use ambii.

Does ambele coate change its form because it follows the preposition pe?

The preposition pe requires the accusative case. However, for neuter and feminine nouns (and their modifiers) in the plural, the accusative form is identical to the nominative (dictionary) form, so ambele coate does not change its spelling here.

Why does the vowel change from 'o' in cot to 'oa' in coate?

This is a regular sound shift in Romanian. When the plural ending -e is added to a neuter or feminine noun, it frequently triggers the vowel o in the root to open up into the diphthong oa. You will see this same pattern in other words too.