Recently I learned how to add my own widgets to the tooltray, but I'd also like to remove the default widgets (eyes and clock) to further customize my desktop. Is there a way to do it?

Widgets are stored in /system/tooltray. If you delete eyes or clock from there, they would theoretically disappear if you reboot picotron, but system will also regenerate upon reboot. You can localize your system to make permanent changes to it, easily done with localsys (load #localsys and follow its instructions).