If the shade you chose has a 6 in it, i.e 10/36, you won't need to add violet. 6 is violet, so any tint with a 6 after the / has violet.
The first number is the level or depth. DD looks to be a 10 in the pic you showed, which means you need to stay with a 10/something, 9/something or 8/something. If you go lower than this, you need to 'pre-pig' with the second tint I linked (the perfecton one) to replace the warmth as Lilac mentioned and prevent more green.
For a level 6 you'd pre-pig with orange, then apply the level 6 tint over the orange hair. For level five you need red/orange, lower than that you need red.
If the tint you choose does not have /6 in it, there is a possibility the brassiness would show through, so you would pre-tone (with prefecton). It's mixed with water, applied to towel dried hair and not rinsed out and you throw on the tint over the top.
They're designed for home use for use between pricey colour services or to tone severely compromised hair so you really can't go wrong with them. You apply to damp hair, comb through, leave for 20 minutes and rinse off.
There really isn't a way to make them last longer, sadly. They are a true-semi permanent shade, however a little goes a long way, so one bottle should give you a couple of applications. Other than that the advise would be usual for coloured hair - use a colour safe shampoo (not violet - they tend to be quite harsh), wash your hair in cool water and go easy on the heat tools. They'll still rinse out within 4-8ish washes.
My DD uses them as she does have severely compromised hair, after also playing with bleach. They wash off her hair within a week but she washes her hair daily and it is extremely porous.
You don't need to apply at different times for different sections but if you find the ends are fading before the mid lengths, you could re-tint the ends and leave the mid lengths, it's not at all damaging to the hair and can be used as often as needed.
I'd go with the 10/36 if DD is happy with that, that way you only need to apply the tint and not bother with any pre-pigging or pre-toning.
As an aside and for future reference 'dyes' are normally used as toners these days, so if it's suggested in the future don't panic. They're applied differently when used as a toner and would last longer than a true semi. I can't understand why the stylist didn't tone with a quasi dye to begin with? DD's hair does not look severely compromised in the pics, unless she's hiding an inch of split ends under leave-in conditioner?
If the salon is offering to re tone for free, let them and once that fades then you can try another salon or buy a home toner, although 'lifting' screams bleach bath to me, don't let her lift anything without explaining what she is doing first i.e pre-toning with red to neutralise the green = fine. Bleach bath/soap cap/anything involving lightener = not needed. Bleaching out the green would be overkill, it'll be gone in a few washes.
If you want something longer lasting ask the first salon to tone with a violet based quasi colour when/if you go back to have it corrected. If they refuse, ask them to explain why. The only time I wouldn't use a quasi to tone with is if the hair was at it's limit and using an oxidative tint would be likely to cause breakage.