Learn Farsi: ما نه دوست ایرانی هستیم.

In Farsi, whenever a noun follows a number, it must be in its singular form. Even though the meaning is plural, you literally say "nine friend" (نه دوست) instead of "nine friends".

Isn't the word نه also the word for "no"?

Yes, they are spelled exactly the same! However, the number nine is usually pronounced noh, while the word for "no" is pronounced na. You will easily be able to tell them apart based on the context of the sentence.

How do the number, noun, and adjective connect here? Is there an ezafe sound?

Numbers like نه come right before the noun and do not need an ezafe (the short "e" linking sound) to connect to it. However, the adjective ایرانی comes after the noun, so it does need the ezafe link. When spoken, it sounds like noh dust-e irani.

Since دوست is singular in this sentence, why is the verb هستیم (we are) plural?

The verb always agrees with the main subject of the sentence. The subject here is ما (we), which is plural. Even though Farsi rules force the noun to be singular after a number, the people being talked about ("we") are plural, so you use the plural verb هستیم.