What is your natural hair colour?
If it's black, you'll probably need 40 vol. But as you only want to dye it over with golden brown/copper auburn you won't need to leave it on as long as you would if you were going blonde.
I would say if you bleach til the hair is a yellow-orange colour, the colour you want should take ok then.
Although, it might take a bit longer bleach as you have the blue-black rinse in. And you might get some greenish inbetween stages. Maybe wait til most of the rinse has washed out?
When I bleached, most of my hair was relaxed. The damage wasn't that bad until I started constantly using direct heat and getting lazy with the deep conditioning.
Now, what's left of the bleached part is mostly natural (I'm transitioning) and in great condition despite the bleach!
So, double processing IS bad, but damage can be reduced by not using direct heat, deep conditioning and generally looking after your hair.