Learn Finnish: Kuinka kallis tämä leikkaus on? - How expensive is this haircut?

Yes, you can! Miten kallis tämä leikkaus on? is also perfectly natural. While kuinka is traditionally used for degree (how much) and miten for manner (in what way), in modern spoken Finnish they are often completely interchangeable when asking "how + adjective".

Since we are asking about price, could I say kuinka paljon kallis?

No, you should just use kuinka kallis. When asking "how + adjective" in Finnish, you place kuinka directly in front of the adjective, just like in English. You would only use kuinka paljon (how much) if you were modifying a verb, such as in the sentence Kuinka paljon tämä leikkaus maksaa? (How much does this haircut cost?).

I thought leikkaus meant "surgery". Why is it translated as "haircut" here?

Leikkaus literally means "a cut" (from the verb leikata, to cut). Depending on the context, it can mean a medical surgery, a financial budget cut, or a haircut (which is short for the compound word hiustenleikkaus). At a hair salon, it is very common to just say leikkaus.

Why is the verb on at the very end of the sentence?