I just had my surgery with Dr. Gur last Monday!!! Will answer your questions one by one based on experience.
1. I chose the standard package so the technicians did my extractions. I am unsure if he uses the same ones in every surgery but the ones I got were excellent and extracted my donor exactly in the way Dr. Gur said they should (he did not want to go into my lower crown at all). 1 week later everything looks completely fine and you can barely notice anything was done. In general, most people would say that you should have as much doctor involvement as possible but many of the top doctors have technicians do all of or part of the extractions. It really depends on what you're ultimately comfortable with. When the doctor does it you know that you're always getting a consistent result, but if techs do it there's more variance, so it might be worth it to do the VIP package just for the peace of mind, you just have to do your research on what the doctor's results typically are. I knew that I saw virtually no difference in results among people doing standard or VIP, and from how ethical Dr. Gur is I trusted that he would have people who knew what they were doing, that's just me though and if I ever go for a second case where my donor supply is more limited I may want more doctor involvement.
2. Again I chose the standard package so my total cost ended up being 3400 Euro. After the 500 Euro deposit I owed 2900 in total. I paid it in USD, which ended up totaling $3150. They accept USD, EUR, and GBP notes. I had just withdrew it from my local ATMs over a course of a couple of days ahead of time without problem (my bank makes it quite easy to increase withdrawal limits). I had no issues carrying cash in the plane, just put it in a secure place in your carry on.
3. Dr. Gur doesn't provide any head coverings and the post-op instructions are very clear on no hats, although from experience from previous clinics you might be able to get away with a VERY loose fitting bucket hat or surgical cap if you want to risk it, but just know the clinic does not recommend it and in general you should be listening to what your clinic says. I brought my wife with me so it was a bit easier, but I am also super self conscious. It helped that there are a TON of people who are flying home after surgery in Istanbul so you would be far from an outlier. In my experience, I was probably way too worried about how I would be able to fly back with scabs on my head but it was honestly fine and the only people who ended up acknowledging my head were other people who had surgery lol, no one really even stared much.