Kerstina, I'm so sorry for your loss. Agree with Badvoc that it sounds as though you had a lovely relationship with your Dad. I hope, in time, that will bring you some comfort.
Your dad's death sounds so sudden and frightening for you all- no wonder you have felt overwhelmed. Do you know what it is that he died from?
I was very 'lucky' in that my parents' deaths (last year, 10 weeks apart), were expected and, as such, I was able to brace myself for what was to come, and I was able to be with them while they died.
Despite this, I still have regrets- there were things I should have said and done which I didn't, because I was exhausted and overwhelmed and devastated.
What I'm trying to say is that I understand why you are 'beating yourself up' over some aspects of what happened to your dad, but you really have done nothing wrong at all. Your conscience is clear. Bereavement is a terrible, life-changing thing, and guilt and regret is very often a part of it, sadly.
I hope you can be kind to yourself and remind yourself that your dad loved you very much. My heart goes out to you. I miss my dad so very much- we all know how you feel, and I hope that makes you feel a little less lonely in your grief, if nothing else. xxx