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1) A plate holds the switches in place so you can handwire your switch matrix to the controller so it's harder than a PCB, but not hard.
The hard part will be finding one with your layout, unless you have a laser cutter or CNC as one off designs are expensive.
If you're a bit crazy and have lots of time on your hands it's possible to build a numberpad from hot glue using matrix keycaps to get the spacing, but I've done it and it's hard to recommend.
2) I'd go with an ATMega32u4. These are widely used so definitely up to the task, and if you get bored/frustrated writing your own firmware you can always use one that's already available.
The most common recommendation is the Teensy 2.0 as it's tiny and easy to flash pre-made firmwares (you still get to customise your layout etc) but if you're new to the game an Arduino Micro might suit you better as it uses their friendly editor/development environment. These are harder to flash premade firmware to though.
Or the cheapest option is called a Pro Micro which lacks both the friendly flasher and the development environment, but is essentially the same. Might have a few less pins... (I've never used one of these)
3) Custom boards are usually programmable including layers, so each key will do whatever you want and you can swap between layers to do different things.
Running programs and system commands can be done through macros (e.g. one key types "Win+r" to get you the run box, then types the exe path, then enter to run it)
The alternative would be to write your own driver to handle the keyboard input and extras, or configure your firmware to have a standard HID keyboard endpoint as well as another endpoint which sends commands to your custom program. (Please don't ask for more details, I haven't done USB driver development since uni 9 years ago!)
4) A couple of LEDs would be fine to handwire, a full board would not be fun. They're easiest used with each one on it's own pin pin on the microcontroller wired up with an appropriate resistor as in the hundreds of "button and LED" tutorials found online.
A few weeks is a very short deadline but depending what you're planning and where you are it might be possible - I wish you luck