THere's some great advice upthread.
doctors (And dentists actually) - the Wiggles, Top of the Tots. There is a doctor sequence and a dentist sequence in amongst the other songs. Then you just wear the right kit for going to the real dentist and voila. Otherwise, the bag of favourite toys is one good idea. Portable DVD player is another
haircuts/washing/brushing
Haircuts: yes, hairdresser at home. Or just be in the bath with him and give it a trim while he's busy playing at the deep end? Minimal faffing and brushing, just a quick snip (much easier with longer girls' hair, of course). Brushing - if it's short enough, it doesn't need it! washing - don't bother? The natural oils quickly rebalance themselves, and just a rinse with water is enough. And that can be made fun
face washing: who?
nail cutting: while asleep is good. Once a child is showing an interest in a long nail, you can offer to trim it and back off if they say no thank you. In the end they'll be asking for nails to be trimmed, and it's perfectly possible to get to that stage without ever pinning child down against their will.
In all of this (and I speak from bitter experience ) I think that adult anxiety over whether child will be distressed feeds into the child's distress. You can kind of ratchet each other up. So perhaps approaching any of the non-life-threatening things on your list of anxieties with the philosophy that if the hair doesn't get washed today it's not the end of the world, then you might be able to relax enough to help your child relax, or else to be able to back off without the thing becoming an issue?