No. Your wife cannot come back to Qatar on the same Visa, Unless the visa carries a multiple entry (which in most cases is not).

You will get a visa on arrival to UAE but you will need to apply for your wife's visa (either from their embassy here or your home country).

one more thing, if your wife exit's the country you cannot apply for her visa for 3 months unless she was on a tourist visa (if you had applied for family visa, you will have to wait for 3 months to apply again after her exit).