I don't usually post, but I lived (and live--14 year old daughter now) with this. You have my sympaties, but years of managing this problem (and consulting with anyone who would listen) has taught me a few things.
Given the intensity of the reaction it could be that she has a meaningful tactile sensitivity (like how some people can't stand tags in their shirts) and that she is experiencing real discomfort or even pain. You need to deal with it as if you were asking her to get a shot at the Dr.'s every day when she puts on her socks.
Some solutions: First, try to find out what's bothering her. Change the tenor of the discussions from total exasperation to becoming detectives together to solve the problem. Some parameters you need to explore: are the socks "bumpy?" (e.g. if they get a wrinkle does it bother her? if so, she may need to wear socks that are really tight and thin to avoid wrinkles.) Does the seam hurt her toes (and therefore has to be perfectly aligned?) Heels also can be a real problem, so posit that as a culprit. (We used to yell at the socks together because they were BAD, and have ritual throwing outs of socks that didn't work.) Our perfect socks were the tic tac toe socks (seamless, available at Lord and Taylor) turned inside out. Also, she could not wear socks for more than a couple of months--the pilling from washing made them hurt again.
Sometimes when you deal with the socks properly the shoe situation gets better. Just be careful when putting on shoes (buy ones that can open wide) that you don't disturb the sock. Putting on the shoes can become a funny ritual together...If you buy shoes at REI or Nordstrom they let you return them after your child has worn them if they are causing a problem that is discovered only after several wearings.
If the shoes are still a problem some suggestions for that: if she wears UGG's in the winter she can get away with no socks and they have lovely soft fur inside. My daughter was able to wear sandals with no problem (certain kinds of course) and we just surrendered--she wore sandals until snow arrived. You could try some of the sport sandals that are more covered than open. Not optimal, but doable.
Last, build in time for sock and shoe putting on. I know it's hard in the morning, but we always had a good 10 minutes allotted for our daily sock time. Sometimes it took 1 minute (new socks!) and sometimes it took 20 (that was tough). Also, let everyone relevant know the issue--that way it becomes acceptable at school (maybe slippers at school if it's really bad) and playdates for her to take off her socks and shoes...
The key is to remember that her experience of the pain is real, as annoying and frustrating as it seems. This can be opportunity for daily fights, or a time that she learns that you will listen to her concerns (a very useful lesson as they get older). That said, I feel your pain. I wish I had the money spent on socks and shoes (oh, and underwear) back--I think we'd have a house at the beach. Good Luck.