Have you tried running this on different browsers yet? And if you’ve got some security suite or antivirus running in the background, have you actually tried turning it off to see if that's the culprit?
I'm not really sure what you mean by "opening a folder"...
You could just shrink your Microsoft Edge window so you can see both the page and the image at the same time, then just drag and drop it. Or, you can minimize the browser and hover the image over the icon in your taskbar—just don't let go of the mouse button. Once the window pops back up, you can finish the drag into the drop zone...
I just gave it a shot on Broadway and everything seems fine on my end. Make sure you're doing it right: left-click the image you want, drag it over to that "Drop Image Files Here" box, and let go...
Where exactly did you track down those Windows 8.1 64-bit drivers for the MP160? Every time I hit up the official HP DeskJet MP160 site (here) and try to select Windows 8.1 64-bit, it just tells me my OS isn't supported and claims there aren't any drivers available. I managed to find this for the MP210, though. Have you tried running that installer yet?
You really ought to get Verizon to handle that for you. It's their responsibility (assuming you grabbed it on a contract right away, though if you're on a prepaid plan, they might give you some grief if it's been over a year since you bought it)