How to change mode from User mode to System mode for LockerService
When LockerService is enabled all your custom components will operate in user mode. The only way to have your components run in system mode is to disable LockerService.
To disable LockerService you'll need to disable the Critical Update (CRUC). If the CRUC isn't present in your org, you can create a new Developer Edition org that will allow you to toggle the CRUC to test your components with and without LockerService. See the "Activate the Critical Update" section of the release notes.
Which structure should I follow for DOM to work with LockerService in User Mode?
With LockerService on, your components can only access DOM elements that they created. As a starting point I recommend running the Lightning CLI to lint your app and catch any obvious issues.