Hi
Well i love Dubai but don't have a family so would probably have a very different holiday but my friends have been with children and loved it (2 similar ages to yours an a grumpy teenager!).
Yes Dubai is safe and there is plenty to keep them occupied. Many hotels will offer a kids club and apart from that you have Wild Wadi Waterpark and Ski Dubai (an indoor ski centre with snow and a specific children's area with tobogan run and snowballing). In future years there will even be Dubailand - a huge theme park bigger than disney. My friend's children enjoyed the deset safari - you're taken jeep and go dune bashing and then to a desert camp where they have camel rides, belly dancers and so on.
Not into golf myself but on my last trip an old chap who was over for a golf tournament was telling me how good the courses are and he'd definitely go back for a holiday.
Hotels are mainly bed and breakfast but there's a great variety of places to eat. Eating in the unlicensed local restaurants or at shopping malls is fairly cheap, hotels will be more expensive but i think it's the alcohol that bumps up the bill and there'll only be two of you drinking.
I can't remember the exact flight time but i think it was around 7 hours. The airport isn't far from the city but hotels are very spread out so it depends which one you book.
Weather is good in Dubai anytime but July/August when it gets incredibly hot and humid.
Again i tend to pick hotels that don't really suit families but my friends have stayed at Hilton Jumeirah Beach which they said was excellent value but a long way out from the malls, city, etc. They've also stayed at the Jumeirah Beach Hotel which is very geared up for families and also has free entry gate to the waterpark.
I'd check both options as it often varies as to which is cheaper - if there's good flight deals to be had book independently but i'd still price it with a travel agent to see if they're offering any deals (free child places, etc) that would bring the cost down.