Thx for your replies!
For PAL or NTSC generation you are better off using a custom chip.
Any suggestions which don't need many extra components? (A few resistors/diodes is ok.)
Every Arduino project I've built has cost more than that... But I'm including the cost of a case, power supply, switches, connectors, LEDs, and whatever other little electronic parts I've needed.
The $50 is indeed without the case, psu, keyboard, mouse, monitor/tv etc. I am in fact counting on people having a keyboard, phone charger, tv available. Do you think it's still unachievable within this price range?
I'm not sure if the existing bootloader will work for you, but you will need a different (or heavily modified) compiler/IDE.
You are right. As I understand, the bootloader depends on 16MHz crystal to work. This speaks for using a Pi Zero. However I still find the VGA solution by Lucidscience quite attractive.
(Just for example, the Arduino compiler doesn't have functions for keyboard/mouse input, text/graphics display, or file system access.)
I am aware I have to write libraries to make a real keyboard/mouse work and may be limited to a ps2 keyboard. Looking forward to this part 
What are you going to use for an operating system & compiler?
I am not sure if I want a compiler on the Arduino itself. (Note sure how to implement this at the moment.) An interpretive language like TinyBasic is a good starting point (Tiny BASIC | Hackaday) for the OS and internal basis functionality.
Making compiled programs specific for the build is something users can do in the Arduino IDE itself, if I provide a framework specific for the build.