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3 of 4 added info about adbd insecure.

After researching this, it comes down to this:

When Droid Explorer launches, it attempts to start ADB in root mode by running adb root. If adb cannot run as root, you will have limited access to some paths.

Found this question here, which is related: Why can't I get root access from shell?

Update: This should work with Rooted devices running Stock (or somewhat stock) ROMs

There is an app on Google Play called ADBD Insecure by Chainfire. This app lets you run adbd in root mode if your device is rooted when running the devices Stock ROM.

adbd Insecure lets you run adbd in root mode if your device is rooted. (Note that if you are running a custom kernel, it is likely that it already implements this functionality)

If you are running a stock (made by the phone manufacturer) kernel on your device, chances are adbd is running in "secure" mode, even if you are rooted. This app lets you run adbd in "insecure" mode, which gives you root access in "adb shell", allows access to system files and directories through "adb push/pull", and lets you run the "adb remount" command to make your /system partition writable.

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Now, I run CyanogenMod 12 (this should also work with CM11), but I don't know if this can be accomplished with stock rooted ROMs.

These settings are for CyanogenMod
In order to allow ADB root access, I go in to Settings => Developer Options. There you will see Root Access:
Root Access

This needs to be set to either ADB only or Apps and ADB.

Adb only / Apps and ADB

If this is not set, Droid Explorer will not be able to launch as root. When it tries to get the folders and files of certain directories, it will be denied.

If you can, try running adb root from a Windows Console window and see what it returns. It may add some insight to if it is possible to have non-CyanogenMod devices run adb as root.