Learn Portuguese (Portugal): O apelido do aluno também é Costa.

No, in European Portuguese, apelido means "surname" or "last name" (like Costa, Silva, etc.). This is a common false friend! If you want to say "nickname", the word is alcunha.

Why is it do aluno instead of just de aluno?

do is a contraction of the preposition de (of) and the masculine definite article o (the). Because Portuguese doesn't use an apostrophe-'s' for possession, "the student's surname" is phrased as "the surname of the student" (o apelido do aluno).

This lesson is about chamar-se. How would we say "The student is called Costa" using that verb instead of the verb ser (to be)?

You would say O aluno chama-se Costa (literally: The student calls himself Costa). Notice the hyphen: the reflexive pronoun -se attaches to the end of the verb. This is called enclisis, and it is the default position for pronouns in European Portuguese.

Could I put também at the very end of the sentence like we do in English with "too"?

It is much more common to place também (also/too) right before the verb it modifies (também é), or sometimes right after it. Putting it at the very end of the sentence sounds much less natural in Portuguese.