I'm very sorry to hear that you lost your mum such a short time ago - you must still be very devastated and whilst your father's actions have caused a lot of distress; none of us can control how our close family or friends live their lives, even if our opinions differ greatly from their's. In most, all we can do is try and accept the choices they have made and move on. I know this is easier said than done, but your father has made his choice - it may turn out that this relationship fizzles out, or grows stronger but on your part, unless you want to lose you father, there is no point in trying to fight against what he obviously wants. You both need to sit down with your father and in a civilised manner, talk through the percussions that his actions will obviously cause if he moves this woman in. Does he actually know how much of a threat your sister perceives this woman as? & what you are both feeling at the moment? I know this woman is a lot younger than your father, but what do you actually know about her? Very little, I doubt - your anger is blocking any desire to actually wanting to get to know this person. I recall the phrase 'Don't judge a book by it's cover' which we are all guilty of doing at some point or another.
As to the inheritance concerns, this is another area where you will stubbornly have to accept whatever your father decides - to live one's life hoping you will get 'this or that' when someone dies is not to be valuing that person in the present tense and you are only thinking of what you will gain. We cannot use our parents as an assurance that we will property, money, etc - value your father in the here and now, forget about what he owns and what you think you will be due - this isn't loving someone. I am sorry if this sounds philosphical, but at the end of the day, family is all that should matter - money & possessions are irrelevant and you cannot replace a person.