Learn Portuguese (Portugal): O vizinho dele cumprimenta-nos.

In European Portuguese, the default position for object pronouns is after the verb, connected by a hyphen. In this lesson, you are learning about trigger words (like certain adverbs and indefinite pronouns) that pull the pronoun to the front. Because this sentence doesn't have any of those trigger words, the pronoun stays in its default position at the end.

Why does it say o vizinho dele instead of o seu vizinho for "his neighbor"?

Both are correct, but dele (of him) is much more common in everyday spoken Portuguese. Because seu can mean his, her, or your (formal), Portuguese speakers prefer to put dele after the noun to make it perfectly clear that they mean "his."

Why didn't the verb drop its final letter before adding -nos?

You might be remembering the rule where verbs drop an -r, -s, or -z before adding an object pronoun. That rule only applies to the third-person pronouns (-o, -a, -os, -as). For first and second-person pronouns like -nos (us), you simply attach the hyphen and the pronoun directly to the conjugated verb without changing its spelling.

Does cumprimentar mean "to compliment"?

No, this is a classic false friend! Cumprimentar means "to greet" or "to say hello." If you want to give someone a compliment, you would use the verb elogiar.