Whoa, that is a handful of questions. I will try to answer some I can help you with, but many of the questions have already been asked and answered in the forum.
1. Yes, as long as they are unlocked and use GSM network. As for the best one, you will have to check with the mobile provider (there are three: Wataniya, Viva and Zain). I am using Skype most of the time anyway.
2. There is one city to speak of really.
I wouldn't say that crime is a problem here. There are parts of town that are nicer and less so, but even if you don't get a place near to the office, you can get everywhere in 30-40 minutes.
3. There are a couple of large supermarket chains that have an organic food section, that is probably your best bet.
4. This is something I would like to know as well. It is common to have a maid that you can sponsor, but I don't need a full time help, just someone to jump in from time to time.
5. Kuwait is quite family friendly I would say. There are some indoor playgrounds that you have to use during the summer season, but weather permitting there are a lot of open air spaces for children and families.
6. There are a *ton* of fitness clubs here, seems that there is a fetish for these things going on here as they are everywhere. 
7. Avenues mall is the largest one, followed by 360 Mall and a couple of other ones. There is a Mac shop almost in each one of them, so no worries there.
8. No experience with local banks, but you will be able to do it with any of them, just check how much they will charge for it. NBK is the largest bank I believe, and I know that a lot of people use Western Union for international money transfer.
9. Yes.
10. It depends very much on the season etc... I have seen return flights to Vienna for example for as low as $300, but most of the time the average price is around $700-$800 for most European destinations. During the summer the prices skyrocket as locals are fleeing the country. 
11. Keep your knees and shoulders covered and you will be fine. I don't think there is a special rule for children, but maybe someone will be able to tell us more.
12. Ouch. The post delivery is a kind of a problem in Kuwait, as generally the parcels are not delivered at all using the regular postal service.
There are services like Aramex that people use when ordering from abroad, but I haven't beer ordering anything yet so can't comment on that yet.
13. I have heard that medicine can get expensive here but is readily available. However before you bring anything over make sure that it is allowed in Kuwait and that you have the proper paperwork to bring it in. No need to bring diapers and baby stuff with you, you can get it all here.
14. No, but there are similar places that you will find familiar.
15. Absolutely.
16. Yes, traffic is horrendous, and Kuwait is famous for holding the top spot for traffic casualties in the world. That being said, you will get used to it in a week or two (not tot he casualties, the traffic) and if you are careful and keep some common sense you will be fine. I was a bit worried initially, but I am completely comfortable now. It might be worth bringing the car over if you can get a decent price on that, but check what paperwork is involved to do so. Here is some very useful info if you are planning to do so:
17. Just use your common sense and get acquainted with the local customs, but apart from you mentioned, there are not many things that I would find important.
18. Yes, you can get a satellite TV with a ton of channels from . Internet is available but the speed and reliability is not up to par with what you are used to, and is a bit pricey as well. No big deal though.
19. My wife and a baby will hopefully join soon as well. Shout if you need help. Also keep in mind that you will have scorching weather here in August.