I keep dragging actions that I want to delete to the actions trash icon. But they just reappear after an hour or so it seems like. I don't close Photoshop down very often and I'm always exporting large batches from Lightroom.
The only way I can think of that may cause that to happen is that your Photoshop preferences are being refreshed and the the loaded actions preferences file was not updated when you delete actions in a sets and deleted actions sets. Or that you deleted actions in a set did not save the set after deleting the actions and later the set was loaded again which restored the deleted actions. You may want to look and see if your user Id Photoshop Preference file "Actions Palette.psp" is updated when you delete action and action sets. This is the file Photoshop savse loaded, edited and added actions into.
Both Windows and Mac OS likt to hide thing on users. So depending the OS version your using yon need to change system settings so that Finder or Windows explorer can see an display files hidden by default. I use a PC so windows and I use Windows XP and Windows 7. With Windows 7 my user id Photoshop actions preference file full Path is "C:\Users\My User Id\AppData\Roaming\Adobe\Adobe Photoshop CS6\Adobe Photoshop CS6 Settings\Actions Palette.psp" Open finder or windows explorer display the folder its in. Then open Photoshop and delete a loaded action see if the file is updated. And if you delete a action from an action set be sure to save the action set so the action will be deleted from the action set saved on disk. If you delete an action set you may want to delete the set you have save to disk so it gone and can not be loaded in the future.