Learn Italian: Mio cugino ha sempre fortuna.

In Italian, you use the verb avere (to have) for many conditions where English uses "to be". Just like you "have hunger" (avere fame), in Italian you "have luck" (avere fortuna) instead of being lucky.

Why is there no il before mio cugino? Shouldn't it be "il mio cugino"?

Usually, Italian possessives need the article (like il mio libro). But there is a special rule for family members: when referring to a single, unmodified family member (like cugino, madre, fratello), you drop the article. So it is just mio cugino.

Where does sempre (always) usually go in an Italian sentence?

Adverbs of frequency like sempre usually go right after the conjugated verb. Here, it sits directly after ha and before the noun fortuna.

If the cousin were female (mia cugina), would fortuna change to match her gender?

No, it stays exactly the same! Because fortuna (luck) is a noun that the cousin "has", it does not change to match the person who has it. It would just be: Mia cugina ha sempre fortuna.