Hi, good questions. Firstly I don't have kids here, but have a lot of friends who do. Schooling can be an issue if you want the best, places are hotly contended for and not cheap.
Vegetarian /Gluten free no problem, loads of options supermarkets are well stocked and even have a gluten free section.
Beach - some good beaches if you're prepared to drive about 45 mins, we make Friday morning (quietest part of the week) our beach time, lovely beach at Fuwairit, nice sand, clean water about 75Km north of Doha. Worth the drive in my opinion.