Learn Farsi: جوجه‌ها گرسنه هستند، با این حال در لانه می‌مانند. - The chicks are hungry; nevertheless, they stay in the nest.

Literally, bâ means with, in means this, and hâl means state or condition. Together, bâ in hâl translates to with this condition, which we understand as nevertheless or even so. It acts as a coordinator of contrast, showing that the second action happens despite the condition described in the first clause.

Could I use a simpler word like ammâ or vali (but) instead of bâ in hâl?

You could, and the sentence would still be grammatically correct. However, bâ in hâl creates a stronger sense of contrast than a simple but. It heavily emphasizes that staying in the nest is a surprising result given how hungry the chicks are.

Why does the second half of the sentence just say dar lâne mimânand without a word for they?

Farsi is a pro-drop language, meaning you do not need to state the subject pronoun if it is obvious from the verb ending. The suffix -and on the verb mimânand already means they stay, referring back to the chicks (juje-hâ), so adding the pronoun ânhâ (they) is unnecessary.

Is it possible to attach the to be verb directly to the adjective here, rather than using the full word hastand?

Yes, in everyday spoken Farsi, you will often hear the verb attached as an enclitic, making it gorosne-and. Using the full, separate verb hastand as seen here is also perfectly correct but gives the sentence a slightly more formal or written tone.