Agree with Ledkr. Can you sit down with DSS, DP and you, and set out ground rules, and consequences, and privileges that will be awarded for sticking to those rules?
I have a similar situation - am here before and after school with my DSs and DSSs. In the beginning it was a nightmare, fighting, winding eachother up. My DS would wind up, DSS would hit out and there would often be fisticuffs between the two of them. It was hugely stressful, and I used to wake up in the morning dreading the next hour.
It has very much calmed down now, with time.
My DP did used to deal with DSS when he got home, and made it very clear it wasn't acceptable to shout at me, or hit anyone else. I did the same with my DS - not acceptable to wind others up to deliberately to the point at which they want to hit you. It was very depressing at times, and I know DP went through a time when he didn't want to come home "just to tell the kids off" - a bit like my waking up dreading that hour! But he did do it, we persevered, and now it is a million times better - I think because all the kids know what the expectations are, and that DP and I will support each other and show a united front.
I think you should have very firm, and very clear consequences for lying and fighting. I would also try to remain calm, my situation has got loads better since I decided I would try my damndest not to shout and come down to the same level as them! I do sometimes leave them to sort out a disagreement too, I found if I always waded in, I often made it worse, and sometimes they can come up with their own solutions eg. one will walk away, they will agree on how to share.
As your DSS is living with you full time, I think you do need to be involved in the discipline. And your DP must back you up. And the same rules must apply to everyone.
I found it helped to write the special rules for the morning down. We all signed it. They are simple, e.g. get dressed, eat breakfast, get bag ready, clean teeth without jostling and so on. The reward for sticking to them is some TV/computer time. The consequence of breaking them is no TV/computer time. This worked as motivation here, but you'd need to implement whatever motivates your DSS.