Learn Farsi: برادر من باهوش است ولی تنبل نیست.

Yes, you absolutely can. Both vali and ammâ mean 'but' and are interchangeable here. vali is often a bit more common in casual, everyday speech, while ammâ can sound slightly more formal or emphatic, but both are perfectly correct ways to connect these two thoughts.

Where is the word for 'my' between barâdar and man?

In Farsi, possession is shown using a short 'e' sound called the ezâfe rather than a separate translation for 'my'. So it is pronounced barâdar-e man (literally 'brother of me'). Because it is just a short vowel sound linking the words together, it usually isn't written out as a separate letter in the Farsi text.

Does the word bâhoosh literally mean 'with intelligence'?

Yes! It is made of the preposition bâ (meaning 'with') and the noun hoosh (meaning 'intelligence' or 'smarts'). Farsi builds a lot of descriptive adjectives this exact same way.

The English translation says 'but he is not lazy'. Where is the Farsi word for 'he' in the second half?

Farsi frequently drops subject pronouns like oo ('he' or 'she') when the context makes it obvious who we are talking about. Because the first half of the sentence already established we are talking about your brother, you can simply say vali tanbal nist (literally 'but lazy is-not') in the second half.