My experience of WhatsApp Web has been that the images do not auto-load (Chrome on Windows 10 and Windows 7). A heavily compressed thumbnail is loaded and I have to manually click each image that I want to view.

I haven't analysed any traffic and I'm not a developer but my layman's guess would be that the WhatsApp mobile app has a pre-prepared Zip file (or some other compressed container) with the 20 most recent messages from each chat plus these compressed image thumbnails and the Web Client pulls that data from the phone upon initialising. I doubt it is a big data transfer.

I hadn't, but now I have.

Using mitmproxy to snoop my phone, I see that all the data coming into my computer when I open up Whatsapp Web is originating from my phone. Including pictures.

That being said, all of my current conversations are on the new "end-to-end-encryption" thing, which might make a difference.

Images and videos are stored on their server for a fixed amount of time, and deleted afterwards. (So you just send an url and the client transforms it into a thumbnail).

You can test this by trying to download an image for the first time after a few weeks, it will tell you that the image is no longer available, and you should ask the other recipient to send it again.

I can't answer that question but I can say that images shared with whatsapp are compressed to < 100kb

I think you should try web app disabling the internet on your phone. Have you tried it? If watsup store the messages on its servers, it would be able to retrieve. But this is also not reliable since they may check phone connectivity before they retrieve from their "servers". I am still skeptical that they may store on servers behind the scene at least for speed and performance.