Unable to delete photos from iPhone 5 Model MD655LL/A

Question

My wife needs to free up some space on her iPhone so she can take more photos and videos.  She has an iPhone 5 Model MD655LL/A.  In General > About the device shows a Capacity of 12.7 GB and Available 1.1 GB.  She has 9 videos and 1,488 photos on this phone. She also pays about $4 a month for iCloud storage.
I've connected this device to our Mac and to one of our PCs.  We are unable to remove the photos from this phone.  Upon selecting a photo from any collection, I can see the trash can icon on the lower right, but it is greyed out.
Now, I've read that Apple makes it impossible to delete the photos that are synced to the iCloud.  This makes sense, since the syncing process would add the photo back to your device if it's not detected in the cloud.  Our dilemma is this.  She needs additional space on her phone. She thought that the iCloud service would provide her with additional space beyond the limitations of her phone where she could store photos without using up her phone's storage space.  Based on what I briefly read, it appears that we misunderstood the iCloud storage, at least in this regard.
Here is what I need to be able to do.
1. Back up the photos from her phone to one of our devices.  Either the Mac or the PC, it makes no difference to me.
2. Then delete the photos from her phone, but retain them on the device where we backed them up.
I expected to be able to use Windows Explorer on the PC in Windows 7.  But upon connecting the phone to the PC, there is no directory to the phone that shows me the image contents.  Instead, there is a folder under "Computer" named "iCloud Photos". Selecting that shows a message where I expected to see a listing of contents on the device, which reads "iCloud Photos is not enabled" followed by a link "Open iCloud Control Panel".  Selecting that link opens the iCloud dialog, showing iCloud Storage of 182.53 GB of 200 GB available.
Selecting the checkbox next to Photos and choosing the Options button just shows a dialog that lets me choose the iCloud Photos Location on my PC.  But I don't see a list of the photos.
3.  Where is a dialog that I can use to drag-and-drop in Windows 7 from the phone to a directory using Windows Explorer?
I need detailed, step-by-step instructions for this.  Do we just need to cancel the iCloud account?  The storage doesn't do us any good if upon deleting a photo from the iCloud the syncing process will delete photos from our computers and our phone.  Do we need to configure the iCloud account differently?  Also, how can we tell if a photo is locked in the cloud, and thereby permanently affixed to her iPhone.
Thanks for helping us.  This has been very frustrating.

Answer 1

The "delete after Import" message will not always appear after the import.  It appears, if you are importing from a camera or card, but I do not see it, when I import from a mobile device with iCloud Photo Library enabled and into a System Photo Library, that also uses iCloud Photo Library.
See the Photos Help:  https://help.apple.com/photos/mac/1.0/?lang=en#/pht6c803201
If a message asks whether you want to delete or keep the photos on your camera after they’re imported, click Delete Items to remove the photos, or click Keep Items to keep them on the camera.
Unfortunately, this Help text does not state exactly, when it will be possible to delete after import and when not.