Determine type of filtering applied between two images
You can use Discrete Fourier transform. One of it's interesting qualities is that convolution in the time domain is translated into multiplication in the frequency domain. 

Applying a filter, represented by the function $h(t)$, on a signal\image $x(t)$ is equivalent, in fact, to convolving the two in the time domain. You can learn more about convolution here. We will denote the output of the operation $y(t)$, and it will be represented by $y(t)=x(t)*h(t)$. Note that $*$ in this context is a convolution operation.

If we will apply a Discrete Fourier transform $\mathcal F$ on both sides we get: 
$$\mathcal F[y(t)]=\mathcal F[x(t)*h(t)]=\mathcal F[x(t)]\cdot \mathcal F[h(t)]$$
Where the $\cdot$ denotes multiplication. This is again derived from the fact that convolution in the time domain is translated into multiplication in the frequency domain.

You can now use simple math to get:
$$\mathcal F[h(t)]=\frac{\mathcal F[y(t)]}{\mathcal F[x(t)]}$$
And hence:
$$h(t)=\mathcal F^{-1}\left[\frac{\mathcal F[y(t)]}{\mathcal F[x(t)]}\right]$$ where $\mathcal F^{-1}$ represent the inverse of DFT.

You can now use $h(t)$ for any new $x'(t)$ by applying:
$$y'(t)=x'(t)*h(t)\implies \mathcal F[y'(t)]=\mathcal F[x'(t)]\cdot \mathcal F[h(t)]$$

Sure DFT is a bit more complex on an image and you will have to learn how to apply it, but the principle is the same.