By over time do you mean the lifetime of that animal or over time as in evolution?
I think it depends on how intelligent and adaptable the animal is as to how much it's behavior can be modified. Certainly environment plays a large part in how an animal acts and reacts. But I think you are confusing the two, one is innate behavior, the other is learned behavior.
You can most definitely breed animals for a certain set of behavioral characteristics and have them breed true. That's how we've been able to produce hunting dogs, guard dogs and lap dogs all out of the same basic genetic material, by choosing the particular behaviors that we wish to reproduce.
I don't think you can really equate dogs and chickens, or even say parrots and chickens though. Chickens are a more basic animal and don't necessarily possess the variety and multilayered behaviors you might see in some higher animals. And being prey animals they might be effected by their environment to an even greater extent, since it's probably more critical to their survival