Hi Kelly,
I've just made the move from Sydney myself and have all the answers! I think!

1. When we get off the plane, when we first arrive in Dubai, do I need to have a car seat for my baby? Do you arrange for a taxi to have a car seat? Or do you just hold on to them with dear life?
I actually have no idea. Taxis as far as I know do not have baby seats. What hotel are you staying at when you arrive? They should be able to organise a pickup for you and also a baby seat if required. We got picked up by my hubby's company on arrival.

2. The company is located in Wafi Residence. What apartment options do we have close to this that is baby friendly? I don't want you to tell me what is nice etc I just need options so I can investigate them. Can we live in Wafi Residences?
You can live in Wafi residences if you are rich. I work in Wafi too actually. I don't think anywhere around Wafi or Health Care City is particularly baby friendly. The family friendly areas are further down the coast to be honest. This whole area is known as Old Dubai, cos it's old and run down for the most part. Wafi and HealthCare city are exceptions to this cos they are new. Areas around Wafi include Oud Metha, Health Care City, Karama, Umm Hurair, Deira and Bur Dubai. The only thing I can compare these areas to in Sydney is Haymarket. But instead of being Chinatown, it's Indiatown.

3. How do I get around with a baby that needs a car seat as I will have to rely on public transport?

Why will you have to rely on public transport? Just so you know it is NOTHING like home. It's packed to the rafters full of poor workers who can't afford cars, and they have a womens section at the front of the bus cos they men leer at you.
If you can't afford a car, you can't afford to come to Dubai. Seriously. Especially with a baby.

4. We are from Australia, so can someone tell me if we need to and if so how we get documents notarised? Do we have to go to Sydney or can we get this done by our local solicitor etc?

You have to get everything notarised, authenticated, blah blah!
Here is your checklist
1) get 30 passport photos, of each of you.
2) Get 30 copies of your passport itself.
3) Get an international drivers licence so you can drive while your residency is being processed
3) Get a copy of your marriage certificate - one to leave at home and one to bring to Dubai. Get the certificate you are taking to Dubai and do the following;
a) Take your marriage certificate to the Passports Office located just next to central station. You lodge it there - allow 5 days to attest.
b) You then have to mail your attested marriage certificate to the UAE consulate in Canberra - you cannot physically drive there and take it in yourself, you must mail it. Allow yourself another 5 days for them to authenticate it.

Then once you get to Dubai, you have to authenticate your marriage certificate AGAIN with the local authorities but your hubbys company will do this (i think).

That's just for marriage certificates.

If either of you have degrees, they must be authenticated as well, using the same process as above. The ony difference is if either of you went to a non state college (ie APM or another college) and not to a state funded university, you have to firstly get that certificate notarised by a lawyer. Then you continue on the process above. I had to get my HSC authenticated in order to work here!

If you have any other questions just ask...