"I read that it is best to work your way up the arm as you try to hit a vein because you can't inject medicines into a part of the vein that is downstream of the blown section of the vein or the medicine will infiltrate. "
Theoretically yes it can. You might want to give the vein some time to heal up before trying a site downstream from the blown site. In reality I sometimes close an eye after about 15 mins.
=====================
"Also, is it possible to blow a vein so many times while learning how to hit a vein on yourself that the veins become permanently damaged to the point of not being able to inject anything into them anymore? "
Theoretically yes, the veins can become scarred and fibrotic with multiple punctures. What usually happens is that the vessel wall weakens and bulges out, making it easier for injection. I doubt you'll be doing so many punctures to reach that stage.
=================
"I realized it might be hard to use the vacutainer holder if I am doing it with one hand since I have to push the vacutainer tube into the holder which is directly connected to the needle . This movement would be hard to do smoothly with one hand and I might push the needle through the vein on accident. Does this blood collection set also work?"
Yes what you mentioned is very common in beginners. Yes, the butterfly needle you mentioned is better for starting out. Cons: this system depends on the pre-made vacuum from the vacutainer.
The alternative is to buy a normal butterfly needle like this and connect it to a syringe and pull the plunger once you see a flashback. Cons: This system is probably slightly harder to visualise a flashback as there is no existing vacuum.
1
It's like the piano.. gotta do it until you're perfect.
If you're just using the normal citrate tube to make prp it's still ok but I would consider that an imperfect solution. A Venezuelan doc once taught me how to make "third world" prp using that technqiue but that's beyond the scope of this forum.