Determining the type of filter on an image
As you said, you can simply plot the frequency response of your filter from $-\pi$ to $+\pi$. 

However since this is a filter only has real co-efficients, you can also just plot the frequency response from $0$ to $\pi$, as this will give you all the information you need. 

This is because for a real signal, the DFT is conjugate symmetric, meaning that all positive frequencies are mirror images of the negative frequencies. In other words, you do not gain any additional knowledge by looking at both negative and positive frequencies. You just need to look at one of them. (Either negative or positive). 

If we do that, we see that your filter looks like so. Here I have shown all positive frequencies, from $0$ to $\pi$. This will give all the information you need. 


As you can see, it looks like your filter is a 'graceful high pass filter'. We can clearly see that it passes high frequencies, and clearly see that it nulls low ones, esp DC. In fact, the sum of all your filter co-efficients is 0, meaning we expect it to null out DC completely. 

Further judging by context, this appears to be an edge detection filter. Edges are characterized by sharp transitions in space/time, meaning high frequencies in temporal/spatial frequency. 

Such a filter would isolate edges.