How to attach wires on a keyboard controller PCB
A keyboard controller tend to use a matrix of connections and scans through them at some interval. Without knowing the setup of the matrix and the controllers behavior, attempting to get the specific combination of 2 individual and 2 combined keys you want is not trivial.
You would be better off using 3 individual keys (say 1 2 3) and use a software re-mapper or hot key or macro tool to change those to f10 f12 and f10+f12 scan codes.
Or use an arduino or digistump or similar usb enabled microcontroller with usb-hid keyboard to provide what you want without all these wiring guess work. At under a dollar for a complete board and plenty of examples online, its dead simple.