Thank you for contacting Telerik support team. I succeeded in reproducing the issue that you described. I recorded a video with my steps. Here is a link to the video:
I dropped 2 image widgets on a page. For the first widget, I chose an image, set custom size and checked "Clicking the resized image opens the image in its original size", then I pressed Save button. I reopened the widget to see that the checkbox is not checked. This is a bug. For the second image widget, I chose an image, set some custom size, did not check the option for resizing and pressed Save button. After that I published the page and viewed it. The first image was able to be opened and resized to the original size. The second image was not able to be opened and resized. So even though the checkbox "Clicking the resized image opens the image in its original size" does not persist its value, the images are resized correct on a public page.
I logged a PITS issue for this bug and also I logged a PITS issue for your feature request about choosing a small image and another big image to be displayed when the small one is clicked.
Here is a link to the PITS issue for the bug:
PITS issue for the feature request: