Everything you describe is so fine. Grief is one of the weirdest things in the world. There is nothing you should do, don't fight it dont deny the pain because itll come back worse. Cry whenever you want to cry. 

The one thing that I want to say is that talking about her will help so much. Tell your dad your sad talk about her with your friends whatever. All I know is that making it all real and accepting the fact she's gone is gonna be the hardest thing to do but it's the best thing you can do for yourself .

You'll never get over it, you'll carry the loss for the rest of your life. Every human that's ever lost someone they loved will too. Even though your dad has moved on doesn't mean he didn't love her or anything. The fact that he is moving on is because he is doing what she would want and that is for you all to be happy .