I haven't been in the same situation, but I can tell you I've had to deal with birthdays and anniversaries and things and they can be difficult (also with an unsupportive mother), so I thought I'd give you some advice.
Firstly - be gentle with yourself. Don't expect too much on the day from yourself, don't do anything you think is going to trigger you to do something that will harm yourself. If crying does help you, then use that as a tool to express how you're feeling.
With all this emotion inside of you, maybe you could write a letter? It doesn't have to be seen by anyone, but if you put yourself in a safe place and just write your thoughts, your feelings, it might help you to get some of it off your chest in a healthy manner, instead of being bottled up.
Keep yourself safe -prior to any urges, remove anything that you could use to hurt yourself with if you can, and put them in a box of something and put it somewhere that's sort of out of reach - you can obviously still get to them, but you'd have to search and reach a lot and while you're doing that, stop and think to yourself -
is there something more productive I could be doing for myelf? Because loss and grief is a hard process, but cutting or purging isn't going to make you feel any better.
It's all about self care of you. Maybe have a moment where you can spend some time thinking of him, crying if you need to - but
plan some things for your day that you can enjoy. Even if it's just playing in the
RYL Games Arcade for a while, to distract yourself even, just try and keep yourself safe and remember we're here for you.
Best Wishes
Aimee xoxo