What a lot of pressure you are being put under!!
I think what concerns me most is that both your parents are expecting you to bail the entire family out, and that doesn't seem fair to me. You are a sibling, not a parent and its not your place to be panicking about where your family live..that really needs to be sorted out between your real Mum and Dad, especially in the case of the 16 year old sister. Of course you can be concerned, any loving sister would be, but at the end of the day you should not have to shoulder the burden of where your family live.
Its a tricky one, as blood is thicker than water, but you do have your own life to lead with your bloke and new flat.
Can your elder sister and other 2 siblings not sort anything out with the mortgage? I am sure HSBC bank where offering a scheme a while back in which you could buy with 2 or more friends, i'm sure this could work with a family as well. Failing that, is putting the house on the market not an option for your Dad? After all, its his problem to sort out. Or does he not want to sell as its his children's legacy?
I don't know what happens to Dad's when the re-marry!! My step mum was the same, she couldn't wait to get rid of me and my brother, we were both out on our ear when we were 18!! Step-monster is the right word!!
I hope you can come to some agreement, but you cant sacrifice your own happiness and financial well being and drop everything for your Dad while he swans off abroad with the new lady!! xx