Learn French: Le bébé a un petit nez. - The baby has a small nose.

You are right that most French adjectives go after the noun. However, short and common adjectives that describe Beauty, Age, Goodness, and Size usually go before the noun. Since petit (small) describes size, it goes right before nez (nose).

Why is the verb just the letter a?

This is the verb avoir (to have). Because Le bébé (the baby) is a third-person singular subject (like "he" or "she"), we use the form a, just like we say il a (he has) or elle a (she has).

Do you pronounce the "z" in nez?

No, the "z" at the end of nez is completely silent. The word is pronounced just like the né part of bébé.

If the baby is a girl, does the word change to la bébé?

No! The word bébé is always masculine (le bébé or un bébé), no matter the actual gender of the child.