Does this panty have elastic at the waist? Is the thread cotton also?
Yes, the waist has a small elastic content otherwise the panties would not stay up:(. I don't believe the thread is cotton - with current production machinery I did not come accross cotton used, especially for underwear.
I did come across Dharma Trading www.dharmatrading.com specifying that the thread is cotton on some of their un-dyed garments but they do not carry underwear.
are the waist elastic and leg opening covered by cotton?
NO, on this style, the elastic is very gentle but not covered with fabric.
You might want to try another style - Organic Cotton Hip Panties, it's very similar though it sits a bit lower but has no separate elastic at openings https://naturalclothingcompany.com/collections/panties-bikinis-boy-shorts-hipsters/products/organic-cotton-hipster-panties