I am sorry but it is very difficult for me to give you the correct information unless I can actually see your hair myself.
All I can do is help you understand and work through the process giving you some facts. If the box color was a true semi permanent red, meaning there are no two things mixed together then that semi colour should be gone, which means you can use hair colour to lift your hair, however it will only lift to a golden brown with a hint of warm copper due to how dark you are naturally. ...More >
Although, you said you still have a hint of red in your hair which leads me to believe the red color was not a semi but a demi which is a colour with a low volume of developer that lasts 4 to 6 weeks, or permanent colour, if that is the case the stylist will have to lift your hair with a lightener (bleach) in order to get the level of lift and shade you are looking for and then tone with a colour.
I am not sure what you mean when you asked if the stylist is ripping you off, you only said he did not feel it necessary to do a strand test as he felt your hair was healthy enough. Also regarding your past experience three years ago of bleaching your hair all over and still having patches of your hair missing confuses me. I would think the patches if you still have them are possibly alopecia usually caused by stress and worry. Bleach can damage the hair if not done correct and if the hair is not healthy enough in the beginning of the process to handle the lifting, but your hair would only break off at the scalp or damaged ends and should have grown back by now as your hair grows 1/2 inch per month and the bleach would only damage the hair on the outside of the scalp not the hair inside.
Anyway I hope this helps in your decision and I am sorry I can not help you more.