Learn Farsi: لباسش پارچه‌ی نازک و آستر دارد.

The noun پارچه (fabric) ends in a silent 'he' (ه). When we link it to the adjective نازک (thin) using an ezafe, we need a 'ye' sound to bridge the vowels. In written Farsi, this is shown by adding the letter ی to the end of the word.

Since we are learning about چه ... چه ... (whether ... or ...), how could we apply that to the objects in this sentence?

You could replace the conjunction و to say لباسش چه پارچه‌ی نازک چه آستر دارد. This translates to 'Her dress has whether thin fabric or a lining.' While this specific combination sounds a bit unusual in English, grammatically in Farsi it perfectly applies the inclusive correlative structure to the two objects.

Does the verb دارد apply to both the fabric and the lining?

Yes, exactly. Farsi places the verb at the very end of the sentence. When you have a list of objects joined by و (and), the final verb دارد (has) acts on all the items in that list.

How would native speakers pronounce the word و (and) in this specific sentence?

In formal Farsi it is pronounced 'va', but in everyday spoken Farsi it becomes a short 'o' sound attached to the previous word. You will hear native speakers say 'nazok-o astar' instead of 'nazok va astar'.