No grace period and no concession for children. If you overstay you pay: QR200 per day. Last month I went through this process because we overstayed by a few hours without renewing the visa.

The visa office next to the airport is open 24hrs so you can extend your visa at anytime during the first month of your stay. After and extension you must leave by the end of the second month. You should not ignore these rules as this is the law of the land.