Will iCloud CalDAV sync my projects and checklists?

Yes. Between your 2Do devices, projects and checklists keep their structure when you sync through iCloud CalDAV.

Other CalDAV apps may show those same items as plain tasks because they do not have 2Do's project and checklist model.

For setup and limits, see Sync with iCloud CalDAV. If you'd rather your tasks also appear inside Apple's Reminders app, that's a separate method — see Sync with Apple Reminders.