I have used a random forest model for prediction of prices. Should the model be predictable in its behavior? By this, I mean that I'm not changing the model and the input, Will the predicted value always be the same?
Yes, the output is the same for the same input.
The "randomness" just concerns the approach how an ensemble is built: bagging (considering random subset of objects) and random selection of features for splitting (when we create each decision tree and choose which feature would be used as a splitting criteria).
Yes. Prediction result of Random Forest model is an average of results of all individual trees, there is no randomness, since you are not changing the trees and result per tree is deterministic.
As, Lana said, randomness is only involved when Random Forest model is being trained.