Another aspect to consider that maybe you haven't thought out is the "sound volume" between cockatiel and a conure.
I'm only pointing this out if you have more sensitive ears or can get more easily annoyed by screeching or loud flock calling, for example.
And about the dog situation, no matter how trained a dog is or how sweet tempered, there is always the risk of the dog to give in to their natural instincts and in the process to attack a smaller pet (in this case a bird) which is viewed as prey. Please check the link below:
For the overall welfare (mental and physical health) and safety of all pets involved, it's always best to not allow a dog (or cat) in the same room when the pet bird is having out of cage time and this is regardless if there is close supervision or not, because at the snap of a finger, things can progress fast and end in tragedy.
We have sadly had many accounts here of beloved pet birds having been attacked by cats and dogs despite the fact that the owners swore their furry companions wouldn't hurt a fly.
If you don't have a safe "dog free" room to take your potential new bird in where he/she can safely spend time out of the cage, then you may need to reconsider on adding a new bird.