So what actually happened was that one child pushed the other and they ended up bashing into a piece of furniture.
In addition, one of them is giving you some lip?
My sister took a CHUNK out of my back when she bit me.
I kicked the crap out of her on many occassions
I played horrible mind games -
I told her santa was dead (on Xmas eve)
I told her I'd spat in her milkshake
I locked her in the airing cupboard
I could go on.
We love each other very much and now, as adults, are the best of friends.
My sons are great friends and play together all the time.
They also fight. Physically fight.
Oh, and teenagers say vile things. The lip some of them can give is staggering. It's not right - but it's pretty much par for the course!
I think that your expectations are unrealistic. Siblings will be bloody vile to each other sometimes. Your children will give you lip sometimes.
Hell, after I'd been told off I would stand at the top of the stairs sticking my fingers up to my dad (behind the bannister!) He would call up - from the living room - "Stop that."
I'd think HOW the hell does he know?!
He knew because he'd been a teenager once too. 
I think the problem here is not so much that your teenager gives you a bit of backchat or that one of your children shoved the other one - it's that you don't see that that isn't abnormal!
Perhaps that is because you don't live with it all the time. I am not having a go. tbh, I really don't care about whether they board or not, that's not my problem and not what you were asking about. BUT. They don't live together full time and you don't experience them together full time - and that's going to have an impact.
You need to understand that what you are seeing is not abnormal or indicative of a bad relationship or that they hate each other or you. You are seeing NORMAL CHILDREN! 
This does not mean you don't deal with it! It just means you don't think there's something wrong with them!