I have come to feel that a lot of doing well at work is acting
- this is not easy when you are terrified.
Look around to see how others are behaving - see if your behaviour fits in with theirs, particularly those senior to you. You want to blend in as much as you can in this situation.
I wonder if you are smiling a lot, partly from being new and wanting to be liked, and partly from fear, and partly because you are a smiley person? I find that other people think that people who smile a lot aren't treating the work urgently enough. Answer the phone seriously, keep a serious face on, it genuinely may help.
Don't let anyone give you a new task to do without being exactly clear about its deadline and its importance, so that you don't drop something urgent for something new but less urgent.
I do think as you get to know the notes, you will get faster - I think having to do everything exactly as a past staff member does it would be awful! Slowly that will improve and you will have your own way of doing things.
Code your to-do list carefully. I use handwritten lists alongside my Outlook diary as I find electronic task lists just aren't flexible enough. I use two codes for each item. AAA for stuff I really have to do today, AA for stuff i should definitely try to do today, A for stuff that is urgent but not so important, B for stuff that is important but not so urgent (so longer-term things for example) and C for stuff it would be nice to get done. I then code again for things that are P for phone, I for internet, E for email, R for reading etc. I also keep a personal todo list running alongside (quite carefully written so that it's not obvious that it's personal stuff - e.g. 'call D re appt 2 weeks' would be to remind myself about making a dentist's appointment). That's not so that I can do personal stuff at work, but it's so that if something personal comes into my head, I can write it down then go straight back to work knowing that it's dealt with and I wont' forget it. That can also be for thoughts that aren't exactly actions
Writing your to do list really can help and that's something you can do out of work if you have to - so that you get in to work and hit the ground running - your aim would be to have your manager or supervisor say 'what about X' and to be able to say 'Oh yes I did that first thing'. That will give you a boost of confidence.
Re your difficult times at home, I really feel for you. There's no way round those distressing emotions and they will affect you - you CAN only do your best.