Rotating and Shifting image doesn't change FFT2
Time shift corresponds to a phase shift in frequency domain, not a frequency shift. Since you display only the magnitude of the 2D FFT before and after the shift, you do not observe the phase shift.

$$f(x) \Rightarrow F(w) \\ f(x-a) \Rightarrow e^{-j2\pi wa} F(w)
$$

Rotation in space corresponds to a rotation in frequency domain as you observe.

For proofs, see these notes on Some Properties of Fourier Transform