Why can FFT only operate on images with specific properties?

• Can FFT only operate on Grayscale images? If Yes, why?

• Can FFT only operate on images with dimensions of power of two? If Yes, why?

The Radix-2 FFT can only be used with signals with $2^n$ samples, yes. The FFT takes advantages of the number of samples in the signal by reordering the signals samples and then taking the fourier transform: Even numbered samples and odd numbered samples. By cutting the size of the transform in half, you only need a quarter of multiplications as opposed to using the DFT on the full signal. But what stops us from reordering the samples again? - Nothing. So we reorder again and again and again until we end up with a "one bit DFT", do the transform and reorder all of the samples recursively. This allows us to keep the transform simple, but it only works with powers of 2.
Using a different Radix, for example 4, the same principle applies (Radix-4 further reduces the number of multiplications compared to Radix-2), but now the signal has to be $4^n$, a power of four: 4, 16, 64, 256, 1024, ...