Browsers are traditionally more geared towards photographs rather than pixel art and by default will attempt to resample an image when the page is zoomed in or out by any amount. The technique is good for photographic images since it help reduce blockiness or make an image appear smoother, but it's a bad thing for our purposes since we want that fine detail preserved.
Best results with screenshots can
usually be achieved by setting the zoom level to the default level which is usually 100%. Browsers are all different, but the most common shortcut for reverting to the default zoom level is Ctrl+0. Some versions of IE will use Ctrl+* instead.
Some browsers have an option to zoom text only and always leave images at a specific zoom level, which might explain why images don't appear to be shifting at all. Perhaps the images are fixed to a zoom only slightly over 100%? That would probably be enough for the browser to say "we're zoomed in, so I'll go ahead and resample everything".
—
Anyway:
You can usually select a particular item in the sprites layer by first disabling the foreground layer entirely (click the button with white text that says FG). Even if there isn't anything in the foreground layer, it being enabled will prevent you from accessing items underneath. With that out of the way, you should be able to right click on an object and select "View Image", "View Background Image" or whatever. It can be a bit tricky with some of those moving items though.
A probably better alternative is to save a local copy of the page. Usually this can be done by right clicking on a page and selecting "Save as...", "Save page as..." and so on. This will bring up a typical save dialogue. If you select the type "Web Page, Complete" or similar, it will save the HTML into a file as well as all the loaded resources (such as images) in a folder next to it. From there you can find all the images from that page in said folder.
The DKC ropes appear to be a
single tile repeated vertically and a separate image for the
bottommost part.