In your taxi transfer if you order a private one there will be some sub standard car seat provided.
Generally speaking the local taxis do not provide car seats and they are the cheapest way of getting around.
Depending on your car seat if it doesn’t fit onto your pram it might be quite difficult to carry around once you’re in the malls? Sometimes there is a drop off station / lockers that you can use but I don’t think a car seat would fit in the lockers, they’re mainly used for shopping.
We found Christmas extremely dull in Dubai, it’s mostly brunches that people do. Depending on where you are staying I could give you more in depth ideas of what is happening around the city.
Ikea do a roast that is really good value, not sure if you have to book for Christmas Day. We ordered a turkey this year just gone but that was for a take away.
Otherwise generally speaking it’s the same as anywhere of what to do with a young toddler. Beach, swimming, soft play (in every mall), you can do a dessert safari but take the car seat. It’s not specifically designed with babies in mind as you would an all inclusive resort aimed at families as it’s considered a city break.
Have a wonderful holiday!