Why did my installer install 1/4 round corners like this and how can I fix it?
That's the worst job I've ever seen at installing quarter round. As others have said, your installer was incredibly lazy.

There are several different techniques that could be used for terminating the corner rounds in both the corners and at the ends.

A simple 45° miter is the simplest for both inside and outside corners. If you want to get fancy, you can cope one of the pieces. Both of these joints look identical when they're installed.



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For the bare ends that stop at a non-corner location, you can do either a mitered return, or just miter the outside edge at 45° and let it angle up against the wall/cabinets.



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As for your questions about fixing the quarter round without buying new, you can probably get away with doing so for your cabinets, but not on the inside corners. For the cabinets, it will look a lot more natural if you miter the ends so they taper back up against the cabinet.

For the corners, you could buy a short length  (12-24" or so) and splice it by cutting back the existing quarter round a slightly smaller distance. Then miter or cope the new piece appropriately.