Can a cockatiel and cockatoo live together in the same cage?

That many birds should get along fine as long as you have enough space for them. Four birds would need a very large flight cage or even an aviary to ensure they all have the right amount of space and room to move around. You will also need many perches, multiple food and water dishes, toys, etc, to keep them from fighting. As training and care are very in-depth questions outside the scope of a single answer here, I'm pointing you toward some helpful articles! https://www.petcoach.co/article/cockatiels-nymphicus-hollandicus-species-profile-coloration/ https://www.petcoach.co/article/basic-bird-care/ https://www.petcoach.co/article/selecting-perches-for-caged-birds-materials-measurement-mai/ https://www.petcoach.co/article/basic-bird-training-how-to-train-your-bird-to-step-up/ https://www.petcoach.co/article/cost-of-owning-a-bird-setup-supplies-and-veterinary-care/

If the cage is large enough for two birds and they are introduced gradually, another female cockatiel could be added. Adding a male will likely lead to mating and egg laying. Also, the possibility exists that the birds will bond with one another and interact less with you.

You cannot differentiate between male and female cocktail externally, a blood test is needed for that. you cannot keep budgies with cocktails together, you run a very high risk that the cocktails will attack and seriously injure the budgies,

Cockatiels can definitely have big personalities. With consistent and training, they can learn songs and may speak though not as good as other parrots. They can certainly bond strongly with you as long as you put in the time and effort.

Cockatiels are approximately twice the size of a budgie. Both budgies and cockatiels are easily tamed, persistence will be required to trach them to talk however. Personally I think you get more interaction from a cockatiel but it really depends on the individual bird and how much time you invest.

Cockatoos are flock animals. The more interaction they receive the happier and healthier they are likely to be. If it's possible to have them on the house with more consistent interaction I think that would be ideal. If it's not possible then make sure they have as much interaction as possible every day.

I think if the new bird would feel comfortable with a companion in his cage then it would be beneficial in helping to tame him. However, if the new bird acts stressed with a cage mate it would be better to leave them separated. Be patient and go slow with the new bird, using lots of treats to help coax him into feeding comfortable with you. It will also help if the new bird watches you interact with your other bird. Best of luck! Thanks for using Petco Pet Education Center, formerly Petcoach.