After installing iOS 7, photos from family members' devices and photos I deleted appeared on my iPhone 5 and iPad, and duplicated. How do I easily delete unwanted photos?

Do you have Photo Stream turned on? That could account for it. You can delete photos from photostream by following this guide: http://support.apple.com/kb/HT5125?viewlocale=en_US&locale=en_US

here is the relevent bit:

On an iOS device using iOS 7 or later:

1. Open the Photos app.
2. Tap the Albums tab.
3. Tap My Photo Stream.
4. Tap Select.
5. Tap one or more photos. Checkmarks denote photos you have selected.
6. Tap the Trash icon. Tap Delete [number] Photos.