I spent the past year in Dubai, and plan to move to Abu Dhabi for the next year(s) starting this fall. I can tell you there are some terrific mamas to be found, but you may need to do a little "digging" to find your community.
Cynthia's got the heat and humidity nailed. There is also the dust. It is a unique and sometimes confusing combination in terms of weather. November through April are fantastic, weather-wise, if you like sunshine every day.
There are a number of natural parenting and hsing groups in the UAE, on various social sites--FB, Yahoo Groups, etc. You'll want to use Google and specific search terms. There are a couple of quite active expat forum sites that can be very helpful with questions about such things as compensation packages, shipping, residential areas and renting, and that sort of thing.
The population is heavily weighted toward expat, but in my experience the transient attitude of many expats, maybe the self-absorption of the work- and party-centered lifestyle of a lot of non-family expats, and the tendency of many nationalities to seek out "their own" presented a real challenge for us. It takes some time, everyone seems to agree.
We spent last summer there, and I decided once was enough. Kids and I returned stateside this summer, and our plan is to be anywhere but the Gulf over summers as long as dh's job keeps him there. We'll return in fall, once temps fall into perhaps the 90s for highs (maybe?). Or low 100s. Lots of people leave over the summers, but there are always families who stay, too. Malls in Dubai try to put on kid-friendly programming, and there are ongoing sport things. Pool swimming and indoor gyms become a staple, and the malls are where people go outside the house. There are several malls, and each one is built and themed differently, to try and keep things fresh. There are cinemas everywhere. Clubs by interest can be found on Meetup.com. There are a number of book clubs and coffee mornings that can serve as a springboard for meeting people.
For me, as a former summer lover and a farmer, it was too much to ask me to adjust to strolling malls as a sanity-maintaining hobby. Over the rest of the year, I was happy to be found running around Al Safa Park and/or walking up and down the public beaches in Jumeira. I brought a bike but wasn't particularly pleased riding it in Dubai traffic. As a family, we tried to get out and see all the Emirates, as well as Oman.
My favorite parts of Dubai are Bur Dubai and Al Ras. I love the old city areas, love to visit the Hindu temple, take an abra across the water, chat with the shopkeepers across the Creek in Al Ras, shop for wholesale scarves, spices, perfumes at the shops. The Metro offers cheap access to lots of the city, and my kids and I enjoy getting a day pass and seeing how much we can do with a day. If you like a broad variety of ethnic food, you will probably not be disappointed. There are a number of kid-specific things to do, including Kidzania, a children's park within the bigger Creek Park, and some little things at Safa Park like rental bikes and little electric boats. Seasonally there's Global Village, which is hard to describe, but not expensive. Jumeira Beach Park has a ladies-only day on Monday, and boys up to age 8 are allowed. I think Safa Park may also have a ladies' day, but I'm not sure. There are several water parks, if you're into that. Al Ain has a nice zoo. Word is Sharjah has a lot of museums for kids, but I have not explored any of that. There is a little "cultural education" area at Shindagha Heritage Village, and the Dubai Museum is cheap and educational (the first time).
There's an organic grocer in Dubai Mall. I also know Life Pharmacy sells many familiar supplement and homeopathic brands, as well as natural toiletries and such. Lots of variety in supermarkets, and we were able to find nearly everything we ever needed to meet needs and/or satisfy cravings. I didn't personally watch TV, so I can't tell you about what's on.
There are all different kinds of residential areas, from grittier, older, "old town" apartments, to gleaming "new downtown" high-rise apartments, to sprawling villas of Emirates Hills/Arabian Ranches/etc., to older villas of Jumeira or Umm Sequim...plus the beach-lifestyle JBR, the airport-flightpath Mirdiff, and more inexpensive (and more bachelor-filled) International City or Discovery Gardens. Some people live in Sharjah for the cheaper rent, and pay for it in commute time.
What sorts of specific questions do you have?
My kids are a little older than yours and were in school this year. Next year, the plan is to use an online school at home.