Sorry to hear your sad news.
Although I always hope that families can reconcile after losing a parent, I do not that this is sadly not always possible.
You need to grieve, but it may take a while to sink in properly. Grief really is a journey, and your feelings change day by day. What you feel now is not how you will feel tommorrow, or next week, but it is important that you do grieve.
As regards your son's birthday, what rubbish timing! I do think that you need to celebrate (know he's probably in bed now), and maybe in the future you can celebrate his birthday, and your mum's life on the anniversary. It's funny how children pull you through sometimes. When my dad died, my overwhelming memory of the funeral etc, was being with my children.
As for your family, I would write to your sister, say how sorry you are, maybe mention a few shared memories and then leave it like that. If you know who is dealing with the funeral, ring the funeral directors direct and explain the situation. They will be tactful.
Personally, I wouldn't get involved in another battle re: money or what happened in the past. Some people have weird ways of going on, and a battle is just going to exhaust you.
Remember your mum for the good times you had, and the love she gave you. My late grandma used to say that we dreamed if her at the times she was thinking about us in the night, so I bet your lovely mum was sending you a message.
Take care. X