Using Convolution as Feature Extraction
Your understanding of the convolution process is correct. However, note that in the convolution the kernel is first mirrored before the dot product. Though, for a symmetric kernel this does not matter.

Understanding the convolution in your way, of finding specific features, is a similar interpretation as that of a Matched filter. Furthermore, using the convolution to find a specific part of the image (e.g. eye), is sometimes called Template matching (but here, usually correlation is used, which is equal to convolution except the kernel is not mirrored).

Another interpretation of the convolution is done via the convolution theorem, which says that in the frequency domain, the frequency response of the filter and the image are multiplied. Essentially, you can then understand the filter to enhance / reduce higher or lower frequencies, (or do more fancy stuff in the frequency domain). With this interpretation, it sometimes becomes more understandable why a certain kernel has a certain effect (e.g. the Gaussian filter is easier to understand as a lowpass than a filter to look for Gaussian shapes in the image).