Learn Arabic (Modern Standard): كم عمرك؟

Arabic doesn't use words for "am," "is," or "are" in simple present-tense sentences. Instead of saying "How much is your age?", you simply place the question word كم right next to the noun عمرك.

How do I pronounce the end of عمرك depending on who I'm talking to?

The core word عمر (age) stays the same, but the "your" suffix attached to the end changes. If you are speaking to a male, the final ك gets a short "a" sound (عمركَ, 'umruka). For a female, it gets a short "i" sound (عمركِ, 'umruki). In everyday spoken Arabic, people often just drop the final vowel and say عمرك ('umruk) for both.

I learned that كم is used for "how many" followed by a singular noun. Why is it used here for "how old"?

While English changes the question word to "How old" for age, Arabic treats age as a quantity. كم is the universal word for asking about amounts, so you are essentially asking for the numerical amount of someone's years.