Windows – Dual monitors with VGA and HDMI on Intel HD Graphics 4600 with correct screen resolution (Windows 7)

It's very unlikely this will work unfortunately. It's a limitation of many graphics cards. Connecting the secondary external output will either not work, disable the primary output or clone both outputs. I'd definitely suggest testing it with another monitor if someone else you know has one available.

Alternatively, you can purchase external usb video cards that would allow for this. I've achieved this using a Toshiba Dynadock before however I have no doubt there are cheaper options. Good luck!

It is physically possible to attach a VGA and HDMI connector at the same time.

Most likely yes. At least there's no reason to believe the contrary.

Would it be possible to:

1. Attach both monitors and extend my desktop to these monitors without using the native laptop screen.

Yes.

2. Attach both monitors and extend my desktop to these monitors, also keeping the native laptop screen active (also Full HD).

Also yes.

I have found little to no information about the Nvidia 940M's capabilities to run multiple monitors through both VGA and HDMI simultaneously.

It doesn't matter. In an NVIDIA Optimus setup (integrated + dedicated GPU) the Intel GPU is always driving outputs. NVIDIA chip is only used to render graphics. As you have confirmed, HD Graphics 4600 is capable of running with 3 monitors attached. Second source here.

Keep in mind that VGA cable won't offer image as crisp as HDMI cable, especially on high-resolution screens. It carries analog signal which is way more prone to interference than digital one and is degraded further by unnecessary digital-to-analog conversion on the output end and analog-to-digital on the input end.

If your laptop has native docking support (proprietary socket at the bottom), you may be able to get one more digital (ie. high-quality) output out of a docking station with no additional performance penalty. You can also buy a USB graphics adapter, but it will give your CPU a hard time and refresh rate may not be so great.