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Windows 7 user here, though access to a Mac (in case that'd be the only way to do this) wouldn't be impossible.
I know I could download iTunes and sync all the photos, but I have nearly 4000 of them and I'd only consider that a last-resort option. What would be simply perfect: Can I just browse them in Windows Explorer on my iPhone? If not Windows Explorer, some App. If I could even get a list of Albums part of my dreams would come true.
Windows 7 should see your connected iPhone as a Photo Camera. Just use the same software that you'd use for importing individual photos from a regular USB-connected camera. Also make sure to have your phone "trust" your computer by turning on the phone and selecting "trust" from the pop-up.
On OS X, you can use the built-in Image Capture utility (or iPhoto or Aperture if you have those).
Can I just browse them in Windows Explorer on my iPhone? If not Windows Explorer, some App. If I could even get a list of Albums part of my dreams would come true.
Yes, as the posters above mentioned. You can also use a tool like iExplorer, which runs on both Mac and PC, to manage your photos selectively. I do this when I don't want to mess with iTunes.
Further, if you just want to sync all of your photos automatically, and manage them afterward, Dropbox does this very nicely with its Camera Upload feature.
Make sure your iPhone is unlocked before you plug it in to your USB, otherwise your PC won't see any files.
In Mac OS X you can easily explore them and transfer the ones you want with iPhoto.
And in Windows you can see the iPhone as a device in "My Computer" and access the pictures that are stored in folders pending from Apple iPhone\Internal Storage\DCIM. They will have names like 823WGTMA, so it's not easy to know where are the concrete pictures that you're looking for.
Apple has a support document covering this topic: iOS: Importing personal photos and videos from iOS devices to your computer
Search "ePhotoTransfer" in app store. It lets you select photos to transfer. The transfer speed is fast at Wi-Fi speed.
If you don't need to transfer videos but only images, use their free version: http://appstore.com/ePhotoTransferFree.
Video: http://youtu.be/lRvwynV4xyE
