No, if the chambers are good. If the chambers are not good then there will be differences. The walls, roof and floor of a chamber can look like mirrors unless well designed, and if there are any reflections then you get the sum of polar diagrams measured from different directions all added together. There are other ways that chambers can measure the wrong polar diagram. A good chamber will minimise all these errors.
No, in ideal case. But if there is any faulty design in the chamber then the result can be changed, du to reflection of signals. Also if the measurement tool (i.e. SMA connector attached with the antenna) is not good then also the result can vary.
For Fixed antenna, the working frequency of that antenna is fixed. For that frequency, lamda is fixed. In that case, the dimension of the chamber should be larger than 2d^2/lamda (far-field condition). In addition, the chamber should not contain any metallic object. If this condition is satisfied (for different chamber), we will always get same far-field patterns.
Due to small dimension of a chamber, the near field data may change for fixed antenna in different chamber.
For wideband antenna, or for multi-mode antenna, take smallest operating frequency. Find the lamda corresponding to that freq. Then apply 2d^2/lamda condition to satisfy the far-field radiation case.