Hello Jferrand526, and congratulations and a warm welcome to the newest addition to your flock. I'm late to this thread, but let me address a few things.
It's too late now, but in future you should always institute a quarantine period when introducing a new bird to your flock. We're talking a 30 day minimum, though proper quarantine is closer to 90. The reasoning behind this is quite simple. You don't know if the new member of your household is harboring any contagious illnesses that might infect the rest of your flock. You should never risk the health of the many for the sake of the one.
But as I said, it's obviously too late to discuss quarantine at this point. I just wanted to advise you in case you get any more birds in the future.
As for Vivian mixing with your conures and cockatiels, there are a few things that you should keep in mind. First, I don't think it's ever a good idea to leave birds of such disparate sizes out and about unattended. Even given the obvious familiarity Vivian had with smaller birds in her previous home, I still don't think it was a good practice. In nature, birds of different species rarely flock together... and never in such close and restricted proximity. And territorial proximity takes up a much larger area. This is why so many birds wind up with missing toes or even feet in captivity. They are being forced to share much smaller confines with other birds than nature ever "intended".
So yes, perhaps the owner from the previous home got away with leaving several different species of birds out and unattended for several years. But even Russian Roulette is good for five turns out of six, right? Having something work for 10 years is no consolation if it ends in tragedy in the 11th. Why take the risk? It's just not worth it. Supervised out time is the way to go.
But there is another factor that comes into play, here. Just because Vivian may have developed a bond of sorts with the other birds at the previous owner's residence doesn't mean that she will necessarily get along with the other birds in your home. Each bird has individual personalities. Just as we do. And just as how you or I may get along with this person or that one, while absolutely detesting yet another, the same holds true in the avian world. No cookie-cutter scenarios, here. So you have to treat each new situation with a bird as something separate and distinct from those that preceded it. Even if Vivian got along with cockatiels and conures in her prior home is no guarantee that she will get along with the cockatiels and conures in your home. A gradual, closely monitored introduction is necessary.
And on another point, is the room for your birds bird-proofed? Are you certain there are no live electrical wires anywhere she may end up chewing? Or paint that she might chew off of a surface and ingest? Or fibers that she might ingest that could lead to compaction? Or threaded fibers she could chew into that could wind up wrapped tightly around her throat and strangle her to death? Nothing metallic (aside from stainless steel) that she could chew on that might flake enough for her to ingest and poison her? No medication laying about anywhere? No cleansing agents or other chemicals? No furniture springs and such she might get to with determined enough chewing? No seat cushions with any gels within? The list goes on and on. It is hard to truly bird-proof a room.
I don't mean to be wet blanket here, but you strike me as a person who has their heart in the right place and wants to know the best way to care for a new flock member. So I just wanted to point out all of the things that jumped out at me as red flags.
All that said, here are a few links that I think you might find helpful in working with Lady Vivian and strengthening your bond with her.
Intro to Eclectus Parrots:
http://www.parrotforums.com/eclectus/52737-so-you-think-you-want-eclectus.html
Bite Avoidance Via Body Language:
http://www.parrotforums.com/training/57935-brainstorming-biting-parrots.html
Bite Pressure Training:
http://www.parrotforums.com/training/63988-bite-pressure-training.html
Target Training: [ame="https://youtu.be/HaOicTtwIZo"]Beginners guide to target training parrots - YouTube[/ame]
Step-up Training:
http://www.parrotforums.com/training/48810-need-help-step-up-training.html#post435993
Tips for Building Trust:
http://www.parrotforums.com/general-parrot-information/49144-tips-bonding-building-trust.html
Eclectus Diet Bible:
fruitandveg