Obviously, you have tried the professional route, and look where its got ya!
1. I think you should use a semi or demi permanent color. Clairol makes both. There would be no ammonia and it would only deposit color, not lift color, so your natural color would not change. Demi would last longer, Semi will not hurt your hair at all.
2. You might want to get two boxes to be on the safe side. Depending on the thickness of your hair, you might be able to get away with one box, I can if I really stretch it!
3. You want an ash color, that will neutralize the red. I assume you still want to see the highlights. A light ash brown will do the trick. (think of red as warm and ash as cool, they will neutralize each other) That will deposit a color over all of your hair, but you won't detect it in your natural color, as it will be lighter than your natural color. On your highlighted area, it will help neutralize the red. You will need to repeat it weekly or so for semi and monthly or so for demi.
You could always do a perm color but once you do that you will have to keep that up, so I would try and avoid that. Did you complain? No colorist should have let you walk away unhappy. Try going back and telling them it is not what you wanted and see if you can get them to fix it for free! (Yea, right!!)