Learn Romanian: Vreau să închiriez o mașină pentru fratele meu.

Yes, a închiria can mean both "to rent" (to pay for use) and "to rent out" (to let someone use for pay). Context is your only clue! Here, getting a car "for my brother" heavily implies you are hiring it for him. If you wanted to unambiguously say "to rent out," you could use the phrase a da în chirie.

Why does fratele have the definite article -le attached if the English just says "my brother"?

In standard Romanian, possessive adjectives like meu require the noun they modify to take the definite article. So you are literally saying "the brother mine." Note: in colloquial, everyday speech, you might hear the contracted form frate-meu, but fratele meu is the standard form used here.

Since the preposition pentru requires the accusative case, why doesn't fratele meu change its ending?

Because the nominative and accusative forms are identical for this noun phrase. It remains fratele meu whether it is the subject or the direct object of a preposition. The phrase only changes if it goes into the genitive or dative case, where it becomes fratelui meu.

Why does the subjunctive form să închiriez end in -iez?