How to know if the noise was removed in an image?
You can measure that with SNR, yes.  What you call 'signal' can get a little fluffy, but you should at least be able to compare two versions of the same image (before and after).  

If your image has a nice homogeneous area in it, you can measure the variance of the pixel values in this region, and compare this before and after - so long as the image doesn't otherwise change.  Do make sure that you account for any scaling or shifting that a filter might be doing (an adaptive median filter shouldn't do either of these if it is implemented properly).  

A common method for testing denoising filters is to start with a pure image (one that you define yourself), then add noise to it artificially then compare the denoised version to the original.  You will need to make sure to add the correct type of noise for your context (is it simple Gaussian, or Poisson, or ...?).