Learn Finnish: Nauratteko te usein?

The dictionary form is nauraa (to laugh), which is a Type 1 verb since it ends in two vowels. To find the stem, you just drop the final -a, leaving naura-. Then, you add the plural "you" ending -tte to get nauratte.

What is the -ko at the end of Nauratteko doing?

That is the question particle! In Finnish, you do not use a helper word like "do" to make a yes/no question. Instead, you attach -ko (or -kö depending on vowel harmony) to the end of the verb. This turns the statement "you laugh" (nauratte) into the question "do you laugh?" (nauratteko).

Is it mandatory to include the word te (you)?

No, it is optional. Because the verb ending -tte already tells us exactly who is doing the action (plural or formal "you"), you could just say Nauratteko usein? However, including te is very common in everyday speech and can add a slight emphasis.

Why is the verb at the very beginning of the sentence?

In Finnish, when you form a yes/no question using the -ko ending, the word with that ending must always be moved to the front of the sentence. Since we are asking a question about the verb, nauratteko gets pulled to the front, pushing the subject te to the second spot.