I'll just chuck my two cents worth in here too!

I have been experiencing the same behavior with DSD, who is also 6.. They sound exactly the same.. The lying and the tantrums and just doing things that she knows she is not allowed to do, or if we ask her to stop doing something, she will try to do it again in some form
Or another to try and get away with it. E.g she poked her tongue out at DH after he told her off for doing something, I told her to cut that out and not to do that again as it is rude, well she just HAD to do it again and when I caught her she pretended she was licking her lips!! Just these little instances and the big tantrums and really exhausting...

We keep her in a very set routine at our house, eg when she gets home from school she has to do homework straight away, then play, then dinner, shower and bed.. If this routine is disrupted or she hasn't had enough sleep the bad behaviors usually comes out..

The other thing we do is set clear boundaries and consequences... If she does this wrong then she goes to her room etc.. So she knows ahead of time what's going to happen and if she chooses to do it anyway she will have to cop the consequences..

I think the real thing is to follow through and make sure she knows the consequences of her actions.. Every single time..

My DSD knows that if she talks back or chucks a tantrum it is straight to her room and that's final, no amount of crying will change that..

With the lying if we KNOW she is lying we make her admit it, there is no blaming anything else, she has to accept responsibility for her action.. Even if it takes an hour in solitary confinement (her bedroom) she will not be allowed out until she tells the truth!

I think in the end it is just a phase, and they will keep pushing you and keep pushing you, but you have to stand strong and set your boundaries and stick to them..

Goodluck!