Learn Polish: Czy ona ma nowe nazwisko? - Does she have a new last name?

Nazwisko means last name or surname. It comes from the same root as the reflexive verb nazywać się (to be called). While you use the verb with the particle się to say what you call yourself, nazwisko is the noun specifically for your family name.

Why is the word for new spelled nowe here instead of nowy?

Polish adjectives must match the gender of the noun they describe. Because nazwisko ends in an -o, it is a neuter noun. Therefore, we use the neuter adjective form nowe instead of the masculine form nowy.

The verb ma (has) usually makes the following noun change its ending. Why didn't nowe nazwisko change?

You are right that the verb mieć (to have) requires the object to be in the Accusative case. While feminine and masculine nouns often change their endings in this case, neuter words like nowe nazwisko are an exception. Their Accusative form looks exactly the same as their basic dictionary form.

What exactly does czy mean at the start of this sentence?

Czy does not have a direct English translation here. It is simply a question particle. Putting czy at the very beginning of a statement is the standard Polish way to turn it into a yes/no question, doing the same job that Does does in English.