While each doc is different, I highly doubt the doc would be doing the blood draws for the allergy tests. Skin testing is more accurate than going by the blood results so you shouldn't be drawing blood on each person who comes through.
To mix vials you just go by a formula, for example, 1 cc from this vial of cat into a 9 cc vial of saline. The formulas are almost always the same pattern so its easy, you just follow the directions. The danger lies in letting your mind wander when doing it and making a mistake, for a mistake with that can have anaphylaxis consequences, especially in multi-user vials. You need to guard against doing that, such as double checking yourself and ensuring that the fluid in the 1cc cat + 9cc saline = the same fluid height as in a 10cc vial. You need to be organized when doing this, and have only the two vials in front of you at a time so that you don't accidentally wonder if you pulled from another vial instead of cat.
Depending on how aggressive the immunotherapy program is you may be dealing with reactions a lot or not much at all. Its important to evaluate each person for safety before giving them their shot.
Yes, I agree with the concern about being solo. A doc should ALWAYS be around if you are doing immunotherapy because the risk for anaphylaxis is very real. The great thing though is that if the program is set up safely, your patients will wait there 30 minutes after their shot, and you will have enough time to monitor them and act immediately if they begin to show the slightest signs they are deteriorating, such as coughing. Almost all of the time you should be able to head off a full blown anaphylaxis before it gets bad.
Concerning the incompetent comment it could stem from so many different things. It could be the doc just being a jerk, it could be that the nurse was not taking mixing seriously and caused a patient harm, it could be that the nurse was not taking proper interventions when it came to anaphylaxis, or that the nurse was not assessing the patients before they got their shots for safety, or it could be many other things.
If you do get offered the job, or are just curious, PM me and I can go into much greater detail about what allergy entails. I've known allergy as a patient at multiple clinics, as well as from a nurse perspective.