When I right click and select Properties on the new icon, there is no Change Icon box anywhere to be found. I see also that this icon Opens With: Windows Photo Gallery. I am wondering if that is the reason I have no function with this icon-it is because this icon is just a hollow image pasted onto my desktop from a photo gallery. Does that make any sense?
Yes, it makes perfect sense. You really just have pictures there - not shortcuts. These are .ico files, not .lnk files (which is what a shortcut file is). The properties for these .ico files are going to apply essentially to picture files and not to shortcut files (which is why the target location and choose icon and other features don't appear there since those only occur with .lnk files (and also sometimes with some program files)).
Yes, I use the Image Icon Converter you have found. It is the one I purchased. I did not see that it was not tested for Vista. Shame on me. However, I have not used it much but it seems to be working fine-at least up front-it seems to do what it says for whatever button you press. Underneath, there may be problems like what I am having. I don't suspect it as being the problem but I don't know.
With it not being listed as compatible with Vista, there's no telling how it might work or not work (or how it might interfere with the system or any of your other programs). If it worked fine with Vista, you can be sure they would list it as being compatible since that's good for marketing and for sales. Perhaps it had something to do with the delayed reaction to working as you wanted. Then again, it may now work fine and we'll never know the reason for the initial problem. There's really no way to know because they do not reveal the reason it isn't listed as compatible (and have no clue why it suddenly started working for no apparent reason). Let's just be thankful it finally worked and hope it continues to do so and without that inexplicable delay.
Upon thinking about the problem now, I wonder if my using CCleaner this morning cleaned out my computer of something that may have been preventing the icons from working?
This is certainly possible, though I'm not sure exactly what it could have cleaned that would have caused that type of problem (and I never personally use it on the registry because the risk is greater than the benefit IMHO). CCleaner is a very good product and used properly can be quite helpful and a good system utility, but be careful using it - if you make the wrong selections you can accidentally delete something very important and cause yourself a major problem (even to the point where only a clean install can repair it).
Thanks for your help. Unless these icons revert, I will consider this problem solved. Not sure how it started working though.
I'd give it a few days or a week or so before marking it solved. To mark it solved, click on the red triangle in the upper right of your post and just type "solved" in the box and submit it. When moderators notice, the thread will be marked with a green check to indicate it has been solved. This helps so others don't come here trying to help only to find it resolved and it helps viewers find problems with solutions (though in this case, we don't really have a solution per se - just that it started working).
We may never know why it had the initial problem, why it started working, or it if will continue to work without any further issues. I don't expect the icons you've converted successfully now will revert in any case (unless you ever restore the icon cache - then I don't know what might happen) - I'm more concerned that that initial problem will return. For now, we can only hope it continues to work without problems and that the issue is truly solved.
Personally, I'm overjoyed to hear it is working - not only because your problem is solved but because I really had no ideas about how to fix the problem beyond what I'd already posted. Now I don't need to worry about researching it any further

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Good luck and best wishes.