You can't understand why this isn't possible unless you look under the cover at the actual technology involved. "But I can plug one side of a cable into the USB port, an the other side into a monitor, and it works, so why can't I do it the other way round?" isn't going to help.
Your USB 3.0 port consists of a number of high-speed serial lines. Normally they would speak the USB 3.0 port protocol, which has nothing to do with graphics whatsoever.
Now some people though it would be nice if you could use the USB 3.0 port for some other purposes, so they invented alternate modes. These use some or all of the high-speed serial links for other purposes, with a different protocol. And just because you have some device somewhere with an USB 3.0 port doesn't mean this is possible.
So there are laptops where you can use the USB 3.0 port to send DisplayPort data: The framebuffer scanning output of the GPU is wired up to the USB 3.0 controller, and the USB 3.0 controller can be switched to output this data on the serial lines, using the DisplayPort protocol.
This means you can plug the other end into a monitor, and it will display the data.
It doesn't mean you can plug in some other DisplayPort data source to your laptop, and your laptop will be able to read the data: The DisplayPort protocol is not symmetric.
If you wanted this, in addition to the extra hardware that allows you to output the GPU framebuffer scanout via the USB 3.0 controller, you need even more hardware that also allows the USB 3.0 controller to read the DisplayPort data, and then you'd have to invent a way to store this data somewhere where the GPU can get at it. And such hardware doesn't exist (at the moment).
So, no, you cannot use an USB 3.0 to read DisplayPort data. Not until someone invents the hardware for it, and puts it in your laptop, and makes the laptop more expensive because of that.
You can buy various other hardware parts that will read various forms of video signals, and transfer that to your computer in various ways, but that will be additional hardware.
1) Yes, you misunderstood. The GPU has parts that read out the frame buffer (called "CRTCs", "Pipes", or other names), and produce a monitor signal. This signal is connected to the USB controller. Not the framebuffer. And because the hardware outputs a signal, it can't input a signal. You can't reverse electronics.
2) The form of the input connector is not important. The signal (and the direction of the signal) is. You can have an input connecter with an USB Plug, a DisplayPort plug, or just some wires hanging around if you fancy doing this. What's important is the hardware behind that: In a monitor, it reads the signal. It couldn't output any signal.
3) A webcam just uses the normal USB port protocol to transfer data. What the computer does with the data is up to the computer: It can save it to a file, or overlay it on the framebuffer so it gets output on the monitor. All this is done by software, and it's fast enough you perceive it as realtime (there's actually a short delay).
Yes, you could have a piece of hardware that reads a DisplayPort/HDMI/VGA signal from a second computer, and is connected in the same way to the first computer as a webcam. You can buy this hardware. See above.