Digital frequency for images
Highest digital spatial frequency is 0.5 cycles/sample. It's related to Nyquist criterion.


  Does it mean that at least two pixels are required to form one cycle?


Yes. Minimum two pixels are required to form the cycle.Try to fit a one cycle of sine wave in minimum number of pixels. You will find that two minimum pixels are needed to fit the sine wave without the loss. Anything less than two pixel you can see the loss.Finally if you fit a full sine wave in single pixel,the pixel is average of whole cycle resulting in total loss.


  Is there any requirement on the values of two samples?


No, there are no requirements like that.

For the edit

The blurred image after Gaussian smoothing is given by $g(x,y) = f(x,y)*h(x,y)$
,where f(x,y) is the input image, h(x,y) is the gaussian smoothing.

In frequency domain G(u,v) = F(u,v)H(u,v)

H(u,v) = G(u,v)/F(u,v)

The Gaussian blurring function h(x,y) = ifft(H(u,v)) .

Using only the cutoff frequency from the FFT of the blurred image g(x,y) you will get the good approximate of your gaussian filter but not accurate one. The reason behind this is the input itself might not contain high frequencies but from the FFT you might think of it caused by the Gaussian smoothing. But for most natural scenarios you can get good approximate of your Gaussian smoothing from cut-off frequency itself.