If this was my little girl, I would let her do virtually anything.
If you decide to go for it:
The earrings.
It is the right decision to make sure you see the doctor first, if her immune system is too weak to fight infections effectively (if this is the case she will probably already be on antibiotics as a precaution) then an infection could spread through her body pretty quickly. Septicaemia could be life threatening.
If her immune system is still weakened, but is not dangerously low, your doctor may okay the ear piercing. If he does, see if the doctor will do it. A lot of people who work in accessory stores or corner jewellers did a one day course or learned to pierce ears from a book.
For her, getting them done in the cleanliness of a doctors office will mean that there is less chance of an infection. A shop, which will quite often do it out in the open with everyone watching, will be full of coughs, sneezes and farts.
Also a lot of the equipment used is shops is "suspect" to say the least. Make sure they use a disposable cartridge system. Many of the "guns" in use can't be cleaned properly, and a dirty gun with traces of other peoples blood on it is not ideal - even if the studs are sterilized.
If your doctor doesn't pierce ears, try to find a pharmacy that will.
I would also recommend you take her to somewhere like Claires tomorrow morning, before you see the doctor, and buy her some clip-ons or magnets, even better, let her choose them. Explain that until the doctor thinks she is well enough, she can't have her ears pierced because she might not be well enough to fight off germs if they get in the holes, but she can have these for now. Explain you will make an appointment to ask the doctor/you will ask him next time you have an appointment.
As you are reluctant to let her pierce them because of her age, it will also be easier for you to get used to seeing her wear clip-ons for a while first.
Also, both of you must be extra-vigilant with the aftercare.
The hair
If she will go for a wig, then that would be the better option, but if you decide on extensions. then take her to a good, reputable (and unfortunately that means expensive) one.
They will be able to advise on whether her is able to take them OK.
In many areas, you don't need experience or qualifications to be a hair dresser, or sometimes about 30 hours worth of nightschool or a 4 day intensive training course. Joanne's corner unisex salon or Barbaras mobile hairdressing probably won't have the experience.
Places like Toni & Guy offer apprenticeships, and until they are totally confident in their students' abilities - they perform their work for free or cheap on 'models'. If you explain why you have come, then you will be given a consultation with someone who really knows what they are talking about - and if it is a bad idea, they will tell you straight.
Give them a call first as you may need to make an appointment.
The last thing she needs is college students who are on their work placement using her as their guinea pig.
If she gets extensions, make sure she knows how to look after them. You don't want them to fall out first time she brushes her hair.