How old is he? Is he castrated yet? Have you had him from a pup?
If he is a puppy then he will grow out of it, given the right training, and its possible to get an adult dog over it quite quickly too, but the whole family has to be in on it. We had an 18month old rottie from battersea who "playbit" i promise you, it wasn't funny! But it didnt take him long to get over it.
When the kids push him away, they are interacting with him and rewarding the behaviour, how old are your children? Are they old enough to understand and take part in a new way of doing things? What i would do was, stand up if i was sitting, walk away, not acknowledging the dog, arms folded so no fingers to snap at. Look anywhere but the dog, direct your attention elsewhere - just keep turning away from him. Harsh as it sounds but we would only interact with the dog on our terms, he could come to us when we called him, not when he decided to dump his ten bloody stone on my lap. He needed to know that being a quiet non pushy dog got him attention and not being in your face. It seriously worked, you could see the wheels whirring in his head when we did this - it would be like, errrr whats going on, i want to play but she wont - oh, i cant make her, because she is in charge and im not, nice dog, result.
So its ignore ignore ignore with the play biting and to a certain degree the jumping up.
Him knocking the children down is a worry, this a dominance behaviour and must be stopped as dogs can get carried away and seriously bite if their "prey" is on the ground. Don't underestimate a labrador, they are a rotweiller without the brown bits! He doesn't sound aggressive but he sounds like he is the top dog and this has to stop. So, no dog on beds, sofas etc, no dog upstairs ever.
If he is not castrated then get him done as soon as possible. Do you ever walk him off lead? how much is he being walked. Jumping up at people is totally unacceptable and can get you into trouble, make sure that whoever is walking him is strong enough to prevent this.
What do you feed him on? Is it high protein rocket fuel? Maybe consider changing his diet to something with a lesser protein content.