1) What are the Public Holidays in Qatar

Eid Al Fitr (approx. 4th week of september)
Eid Al Adha (approx. 4th week of november / 1st week of december)
National Day (Dec 18)

(check calendar with proper authorities for details on days)

2) Are all the restaurant serving Halal food?

Halal food restaurants have a sticker on the door stating so.

If there is no sticker please ask before eating.

Most of restaurants are for sure HALAL.

Regards!!