Is it possible to leave the clock running in order to accumulate money and get research done, without going on missions?

Yes, but it's a very slow process. The better solution is to start a mission, wait a bit and abort it - you won't be punished for it and you will "waste" a lot of time.

Time on map is going real time. like clock on your PC, time in missions is many times faster, but if any country start rebellion you get less money and even lose them. Save only after you completed mission or aborted from debriefing screen because saving and reloading any time remove part of money from you, probably because of hardcoded bug.

I know this is a kinda old of threat but I remember there was a cheat code that would speed up time in the world map window.

Just enter WATCH THE CLOCK as Company name and well, watch the clock...