Can I use liquid nails to hang chair rails in my child's room? What is the average height from the floor that they should be hung?

Liquid nails is fine bit trim nails should be used also. A power nail gun makes a quick job of it.
36 inches is standard but you can adjust for where your chairs hit the wall.

Liquid Nails is some strong stuff but I personally wouldn't use it in this application. If you only used the Liquid Nails, the glus is only adhering to the paint. If someone leans on the rail hard enough the whole piece could come down. Not to mention I'm not sure how you'd keep the rail from sliding down while drying. AND if you ever changed your mind down the road and wanted to remove the rail, you have a lot of patchwork to do. The nails are definitely the right way to go.

If you are painting the railing white and want a real professional look I'd caulk where the rail meets the wall. It gives it a very clean look. I caulk the edges of every piece of trim in my house.

I am with logan on this. Liquid nails is strong but your paint surface is not. By the way, Liquid Nails is not a singular product, they make like 30 different adhesives and each is for a specific application.