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Your dad cheated on your mum, not on you.
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Everything involving humans is more complex and complicated than it might seem at first glance.
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Everybody makes mistakes, even your loved ones.
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You only have one dad, so it's better to forgive them. (I didn't forgive mine for other mistakes, and that was my mistake. Now I'm old and he is dead and that's that.)
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Your mum is being selfish and manipulative because she is afraid and hurting. It's not right what she is doing, but see the points above for her as well.
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Life is hard and unfair and difficult for everybody. For your dad, for your mum and also for you. It sucks when you're stuck in the middle of other people's problems, but remember all of this will pass. And remember to take care of yourself.
Hugs my dude. You'll get through this and so will they.
Edit: 7. Time. Let things take time. Don't rush what you feel or what you should feel. Don't go overthinking everything. Things that are complicated need time to settle.

Go on the date, have fun, spend dad's money. When he asks, tell him it was great and she spent all night pegging you in the arse and her side dude joined in. Then ask him for more money so you can do drugs together.