Favicons (icons for the web pages you bookmark) will disappear when you use Internet Explorer. Fact of life.
The icons are stored in temporary internet files and when you clear your temp files, the favicons (favorites + icons) are dumped too. No way to stop this behavior because Microsoft will not fix it. MS also has not fixed many of the vulnerabilities that kept causing my computer to have headaches when working on the web. They forced me to go to other options.
Firefox does not store the favicons in the same way. When you add a page to your favorites in FireFox, the favicons are stored differently so even when you clear your temp folder, the favicons remain.
As I mentioned before, you might consider downloading FireFox and trying it out. You might end up making it your primary browser as I did. Since IE version 8 instabilities, many of us moved away from IE.
I have IE and still use it for certain sites, but FireFox is my primary browser. Pages load faster for me, I can grab videos, use web developer to "see" how sites are made, and certainly the saved passwords are easier to manage in than in IE too. I got tired of web pages locking up in IE. They don't in FF. Even with the "compatibility mode" IE has difficulty showing old or new web pages. I got tired of popups (even with a popup blocker), don't have them with FF. FF plays well with my photo and web building programs.
Use whichever browser YOU like. It is a personal choice.
There are compromises no matter which browser you use. For me, FF has the least downside, most to gain.
I know Rick also uses FF as primary.
FF Favicons: Bookmarks imported from IE may have missing favicons. Visit the page. Delete the bookmark. Copy the URL to clipboard. Close FF and come back in. Then visit the page and make a fresh bookmark. This time your favicon will stay.