Qatar is perfectly safe for Westerners, but go easy on the cleavage and tell your kids NEVER to kiss or cuddle boy/girl friends in public. Displays of skin or sexual affection are considered offensive and thus can get you into jail. You won't get mugged, though.
There is an Applebees here but they don't sell alcohol: only the hotels do that. Alcohol is easily available from a well-stocked government store to anyone with a Resident's Permit and a booze licence - or who has a friend with same - which is why a lot of western socialising here is done in people's homes.
Minor detail: In July and early August, most western kids and teens will be away for the school Hols, and you won't want to be out exploring cos it's just too darn hot. Life centres on the malls.