The street we have booked an apartment in Madrid is Calle Dos Hermanas, near La Latina metro. Is it safe to walk around in the night with young kids?
Is it walking distance from Plaza Mayor/Puerta del Sol?

La Latina is very safe. I live very close to the metro station and have a small child and have no fears whatsoever--there are always people out on the street. As in any city, just watch your belongings when you are eating, especially outside on a terraza (outdoor cafe) and in big tourist spots like the Plaza Mayor, Puerta del Sol and Gran Vía. Madrid has lots of gifted pickpockets, but it is not at all a dangerous city.

Plaza Mayor and Puerta del Sol are just a short walk away. There's a nice playground in front of the Palace in Plaza del Oriente. And another in Plaza Santa Ana. And many more in the Retiro and sprinkled all around the city.

One of the best things you can do is just to loose yourself in La Latina though... try to make your way to Plaza de la Paja - it's a stunning place for a beer at night, and perfectly safe (keep your bag on your lap always though!)

Perfectly safe, as is most of Madrid. Like everyone else has said, just keep your eye (and hands) on belongings and don't talk to anyone wanting you to "sign a petition". They're pickpocketers.

Is the area around La Latina safe, well for an experience you'll never forget head out with the kids (if there still awake) around 11pm, wander where ever you feel like (just make sure you don't get lost and after a while you notice the most bizarre thing, whole families out enjoying an after dinner stroll (yes I know its after 11) I came across a scene which I could only describe as "church just let out" Grandparents chattering, kids yelling and screaming and being kids, Mothers catching up on the local gossip. Rick Steves talks about it and its really a sight to behold. I took my sister in law their and she couldn't get over the numbers of families out enjoying themselves. Specifically you want to go north of La Latina towards Tiros de Molina into the chinese district, anywhere around there you will find loads of families enjoying an evening.