Young Family/Children in Dubai

We are considering temporarily relocating to Dubai from the USA for my husband's work. With his work we will have housing taken care of and our main concern is how well women and children are treated in the UAE. We are Caucasian and my daughters will be 2 years & 6 months when we arrive.

Are there great parks, activities and museums for the children? Are you able to get out and explore easily? Are there lots of families with young children?

Also, I have read a lot of forums/blogs about non-Muslim women being constantly stared at and/or followed. Is this an issue in your experience?

I appreciate any feedback and thank you in advance.

First, Dubai is a true international city, over 80% of the population is expatriate from 180 countries. There are hundreds of thousands of westerners and many, many families so you won't be unusual! Women and children are treated very well, as you would be in the US and maybe even better.
Yes there are parks, most of the malls have special play areas for children, there are many places to explore, getting around (apart from the traffic problems like any big city) is easy.
Women, Muslim or not, are not generally hassled although in the souk and local shopping areas you may be looked at - there are many men from the sub-Continent here and their culture includes the tendency to stand and stare at just about anything. Overall Dubai is one of the safest places in the world.