I was just wondering if Mac apps can delete files in ~/documents and ~/desktop without user permission?
Applications run with the access rights of the current user (usually), so they can do whatever the user can do in Finder or Terminal as well. This includes creation and deletion of folders and files.
Technically speaking, any app running with user privileges can access/modify/delete user files. This of course, is limited to the owner; meaning an app running as one user cannot interact with files of another user unless the app has "elevated" privileges allowing it to do so.