Gosh - delurking as this post has really resonated with me: back in the mists of time I was your sister, although I was a whisker or two older, down to going off the rails (before and after my mother's death), being handed cash instead of receiving parenting, cries of "for goodness sake she's 16", but most of all the oddness of the new woman (or in my DF's case, in time, women) being presented into my life without so much as a by your leave.
I am very, very sorry for the loss of your mother
and I'd like to reiterate that 12 months is nothing, a blink of an eye in your grief, and losing your mother young is a titan of a thing to deal with. Your DSis acting out isn't something I'd judge her for - it may be the one rational (unhelpful perhaps) way she has of expressing her emotions, it may be one way she gets engagement from your DF.
I wonder though if your sister has the double whammy of having lost both of her parents, in the sense that she once had two parents with some decent parenting (perhaps all of it from your mother) and now she has lost your mum, and your DF is incapable of stepping up, and actually parenting her - and hearing her. Poor sis
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If your DF is like mine (btw, a 'good' man, but one who is walled off from emotions), he may cynically assume that by introducing this woman that she will naturally step into the mothering role, and all will be well again in his and your lives. He is unlikely also to have considered what will happen if his relationship breaks down later, and this woman was to leave you and your sister's lives. Ideally he would recognise that his beginning on a new relationship may be unsettling understatement for his kids, particularly given the recent world-changing loss of their mum, and discuss the boundaries of that with your DSis at least. FWIW I was quite pro- my DF having a new relationship, but it was awkward and confusing and frightening for me at times and we were very much not able to talk about it.
Your father may hear you if you try to talk to him, he may not. Mine wrapped himself in his bereavement like a carapace and I was never able to get him to understand why I felt bad. What I found desperately helpful was when I later found someone (not a DSis, but family member) who was prepared to listen to my badly articulated feelings, didn't judge me for "teen drama" and was able to see that I was a grieving child. I think maybe you can fill some of that role with your sister.
I don't know if that's helpful essay but to reassure you, I did turn out ok! Time helps, and I suspect if you can find a good, grief counsellor(s) for you and your DSis that would help too. I'm doing that only now I'm in my 30s and it is very, very helpful.