Learn Portuguese (Portugal): Mas o apelido não é Silva?

In Portugal, apelido translates to surname or last name. Your first name is your nome. Be careful if you use Brazilian resources: in Brazil, apelido does mean nickname, but in Portugal, a nickname is an alcunha.

Why aren't the words rearranged to make it a question?

In Portuguese, you don't need to change the word order to ask a yes/no question. You just take the normal statement — o apelido não é Silva (the surname isn't Silva) — and raise your pitch at the end when speaking, or add a question mark when writing.

The current lesson is about the verb chamar-se (to be called). Why is the verb ser (to be) used here instead?

You use chamar-se when talking about what a person is called, as in Ele chama-se Silva (He is called Silva). But here, the subject of the sentence isn't a person, it's the noun apelido (surname). When you simply state what a name is, you just use the verb ser (to be), making it o apelido é (the surname is).

Why is mas used at the beginning of the sentence?

Mas means but. It is used here to show surprise or a slight contradiction, just like in English. If you expected a person's surname to be Costa, but you see a different name on a document, you would ask Mas o apelido não é Silva? (But isn't the surname Silva?).