Like Graphista explains we are all different, however, if you happen to have a condition called EDS (Ehlers Danlos Syndrome) as this can cause difficulties with anaesthetics.
Speak with the anaesthetist, write your fears down and ask them to put at the front on your notes so if it's not the same Dr on surgery day it's clear what your fears are and any suggestions to help or make things easier for you and they will be aware and can work with you re this.
They are very highly trained and in my opinion the most responsible of all Drs as they are tasked with looking after you literally at your head on the operating table and making sure you are ok and your vital signs are all within a range, they are the ones who will notice first if anything changes and take actions accordingly.
Im like Graphistas mum, I go too deep and have difficulty coming round,
I also hate the feeling of the mask going over my face so in the past I've had the needle in the back of my hand to count down from 15 to allow a longer time to go out and after the very first time, they made sure I never saw a mask again as didn't put it on until I closed my eyes out, I had short lasting GAs twice a week for 6 weeks so 12 in all, in a short time when I had ECT for depression.
If you have a tendency to get sick easily they can give you something iv to stop that or if your BP fluctuates they will keep a good eye on it, also good to mention at the appointment if you have any allergies to meds, past reactions etc as for example my local hospitals first line in anti sickness meds are Zofran which I always have to turn down as it gives me crushing chest pains and I can't have codeine either as I can't metabolise it, Im a nightmare for a Dr treating me but I have a list uploaded to my records and agreed with my GP of what I can be treated with for sickness, pain etc
I see an anaesthetist for pain management, he did a spinal injection for me in the past just like he was giving an injection in my arm, total ease. He also suggests things to try and see what would help
Their only focus in the operating room is you, anaesthetic and your vitals.
Good luck.