Out of curiosity, why do the favicons of my favorites dissapear after some
days and never come back again?
Favicons will only appear as long as they remain in your browser cache. If
using Internet Explorer, they'll be in the Temporary Internet Files folder.
When the cache is emptied or when the recycling of old files to make room
for new pushes out the favicons, the internet shortcut uses the default IE
icon.
You can grab those favicons out of the Temporary Internet Files folders and
place them into a folder you've created to hang on to them. Rename each one
in a manner that reminds you what site it is for. Then in the properties of
the site's internet shortcut use the Change Icon button to select the saved
icon.
There are also third party programs that collect and manage favicons for
you. An internet search for "favicons" should turn up a few to choose from.