I have a Lumia 635 with Windows 8.1. Any time I download an emailed photo or receive a photo via WhatsApp, I can find all these photos in my "photos" on the phone, a feature I love. My wife has a Lumia 1020 with 8.1. Anytime she downloads an emailed photo to her phone or opens a photo in WhatsApp, the photos do NOT end up in her "photos" section. In fact, we have no idea where they go! We've searched the phone (not via computer) but to no avail. We want to show these photos to others but we re unable. Where can we find these photos?
I'm not sure why your wife's Lumia 1020 doesn't store photos in the Photos section but either way there's an easy way to find them.
There's an app in the Store called Files. It's made by Microsoft. This app gives you a detailed view on your files sortened by type (Documents, Music, Pictures etc). Via this app you can find any file according to its type. Download it and you'll find your file in no time.
Photos downloaded via an Email client will not be saved to phone automatically. They will be stored in the app memory. So they cannot be viewed in photos section. To save a photo from the Mail app follow the instructions in a similar question found here.
However if you are downloading a photo from an Email using a browser your photos will be saved in SDcard->Downloads as said by Nirav Madariya.
To answer your other questions photos saved from Whatsapp will be stored in sdCard->Pictures->Whatsapp folder. You can navigate there using a file manager such as Microsoft's File app.
In the above image you can see the folder created by apps such as Zapya, Shareit and Whatsapp. These apps store images in these folder. These folders are located in SDcard->Pictures