Strange indeed! Are you sure all those websites missing icons have a favicon? I checked tparser.org and it does appear to have a favicon. Would you please try first saving (downloading to a JSON file) your search engines, then removing the extension and the file storage-sync.sqlite from your Firefox Profile folder, and finally reloading your saved search engines in the extension's preferences page and let me know if you get the same results. Alternatively, you may try loading your search engines in a new Firefox profile to see if the problem persists. I'm sorry for the trouble. If the problem remains, then I'd appreciate if you would file a bug on Github. Thanks.
UPD: Would you mind sending me your list of search engines (the JSON file you can download), so I can do some testing on my end (I'm on Mac so I may get different results!)? Also, please feel free to give the latest dev version on Github a try. The code has gone through quite some changes since the last release on AMO. I should be able to release it soon, but I still need to polish a few things and do some testing.