another exercise says "cái ghế" are both "cái" and "chiếc" correct for this one?
I would say they are both correct. The south uses "cái " for almost every object. Cuz we love simplicity. The north is more specific. So they use "chiếc" for chair
Please stay with "cái" and "còn" to make it simple. The North Vietnam has a lot of influence from Chinese that is both good and bad so keep the good ones and forget the bad ones. There are too many articles for Chinese, for example all articles for animals in Vietnamese are "con" but in Chinese cow, horse, rabbit, chicken, pig etc. have totally different articles.
Chiếc, from Chinese 隻 zhī (Cantonese zek3, Hokkien chek); this is a measure word that has different meanings in different Asian languages that borrowed it.