When deleting .zip files the file goes to the recycle bin and reappears where it was deleted from as a zero bit file. the only way to delete it totaly is to shift-DEL. This has been happining for a while now on both my laptops for only .zip files all other files delete normally.
No need to look for any deletion or .zip file related errors. There is nothing like that. This is a completely typical symptom of damaged Windows, with one faulty component unable to remove or change the labels and permissions of files and folders. (Windows does not delete anything, only removes all labels and permissions of files and folders, so the files become invisible to both - to the user and to the OP system, but all files stays on their places). So, you need to fix the faulty OP system and all the other problems will disappear with it itself. Although you haven't said which version of Windows you're running, it's probably been on your computer for a long time and hasn't been maintained regularly. However, maintenance is recommended and also very necessary..
If you want to avoid that in the future and want a healthy system, re-install Windows from scratch. (Format everything!) Check your HDD/SSD now or after your fresh install ! Or mess with the registry and services and use recylce commands, to let this sick patient die slowly.