I mean mental work. Right now I have the coughs and a runny nose, I am taking vitamin C, fluids and all that stuff but there's just a lot of work I want to get done. Some research on some things and other mental work. I would be doing this all day every day, of course with breaks.

So I ask, if I do this, will it pro-long the duration of my sickness? I mean I heard you can't really get "rid" of a sickness or "cure it", as it usually lasts 7 days. So if you can't really cure it then I'm thinking maybe you can't pro-long it as easy neither.?

Enough sleep, keeping warm, plenty of fluids, and humidity are the 'cures' for colds.
I see nothing wrong with working on research. But stay away from other people for their sake, and don't commute through cold wet weather to get there.
Sitting up and walking around, even when sick, is better for you than staying flat on your back, for most conditions. It helps clear your lungs.

I mean mental work. Right now I have the coughs and a runny nose, I am taking vitamin C, fluids and all that stuff but there's just a lot of work I want to get done. Some research on some things and other mental work. I would be doing this all day every day, of course with breaks.

So I ask, if I do this, will it pro-long the duration of my sickness? I mean I heard you can't really get "rid" of a sickness or "cure it", as it usually lasts 7 days. So if you can't really cure it then I'm thinking maybe you can't pro-long it as easy neither.?

There are many ways to look at the situation. Stress can cause your immune system to drop levels and that can prolong an illness. Another thing is that if your in the contagious stages of a flu you can spread it to others. If your running a home office its different then having an office full of coworkers.

So the question is really a situational one.