Learn Turkish: Babaanne çok mutlu. - Grandma is very happy.

Because it does! In Turkish, you specify which side of the family a grandmother is on. Babaanne literally combines baba (father) and anne (mother) to mean "paternal grandmother" (your father's mother). Your mother's mother is called anneanne.

Where is the verb for "is" in this sentence?

There isn't a separate word for it! In Turkish, the "to be" verb (am, is, are) is often just a suffix, and for the third-person ("he", "she", "it"), that suffix is usually left completely blank in everyday speech. Just placing the subject (babaanne) next to the descriptive words (çok mutlu) is all you need to say "Grandma is very happy."

Could I say Babaanne mutlu çok instead?

No, the word order is important here. Just like in English, adverbs that change the intensity of an adjective—such as çok (very)—must be placed immediately before the adjective they are describing.