My wife has an iPhone 6s and I am trying to understand the way in which the device uses local storage. Her photos are very important to her and as such, I want them to be backed up safely. So we pay for extra storage on her iCloud account and her photos go there. Are they also stored on the device? Also, can they be deleted to make more available space on the local device but remain on the iCloud?
I am not very familiar with iOS devices.
iCloud Photo Library is not a backup solution, it's a sync solution.
It can be used as a backup but I don't recommend it. There's other solutions, like Dropbox or BackBlaze, that can serve as an offsite backup (using the 3-2-1 backup strategy).
When you use iCloud for photos, all the photos will be synced to each device signed-in with that iCloud account. Deleting a photo/video on any of those devices will also delete it from all the other devices when they re-sync. See this Apple Support site for more info.
iCloud Photo Library uses your iCloud storage to keep all of your photos and videos up to date across your devices. You can make more space in iCloud when you delete photos and videos that you no longer need from the Photos app on any of your devices.
You can also choose to save space on your device when you turn on Optimize device Storage in Settings. With Optimize Storage on, Photos will automatically free up space when you need it by replacing original photos and videos with device-sized versions. All of your original, full-resolution photos and videos are safe in iCloud and you can download them at any time. Learn more about managing your photo and video storage.
The only way to get a backup is to make a full-size sync to another device and do the backup on that device. Unless your wife has a 128GB iPhone and a relatively small Photos library (say under 20K), the iPhone is going to be set to optimized rather than full-size so there is no way to do a Photos backup with it. Typically, a full-size sync will be done with a Mac so you have lots of backup options at that point. I suspect it can be done with Windows as well but I've never tried it.
A special case would be if your wife only shoots with her iPhone. In that case, the full-size copy of a photo is on the phone until some mysterious process decides that space is running low and swaps it out for an optimized version. If you used a second sync solution like Google Photos, it would capture the full-size version before it was potentially optimized. It's not a true backup since it becomes a separately managed photo archive and changes and deletions to Photos won't get synced into it but it's at least some protection against a catastrophic loss. You also have to be aware of the limits of the free version of Google Photos and if there are any gotchas with videos and Apple-specific formats like Live Photos.
