What forces act on a person sitting in a chair with wheels when they push off another stationary object 
*

*What is slowing me down when I push off my desk?

*What are the "Major" contributors? and

*Does how hard I push with one hand (or with two hands) make a
significant difference?

 A: Friction generated from the rotation of the wheels provides a torque to slow down their rate of rotation. As the entire chair slows down, friction between your body and the seat of the chair slows you down. That's the only major contributor.
A phenomenological approach would be to assume the wheel friction provides a force to slow the chair down proportional to $v^{\alpha} m^\beta$ where $v \equiv dx/dt$ is the velocity of the chair, $m$ is the mass of the Argus + chair system, and both $\alpha$ and $\beta$ are unknown exponents. Let $K$ represent the unknown proportionality constant. Then 
$\begin{equation}
m v \frac{dv}{dx} = - K v^{\alpha} m^\beta
\end{equation}$
Integrating this equation up to the location $d$ where $v = 0$ yields
$\begin{equation}
d = \left(\frac{1}{2 - \alpha} \right) \left (\frac{1}{K} \right) {m^{1 - \beta}} {v_i}^{2 - \alpha}
\end{equation}$
where $v_i$ is the speed you start at from your initial push. You can see that if $\alpha < 2$ then you'll stop after traveling a longer distance if you start out faster. When $\alpha > 2$ you never actually come to rest according to this model, but at a given time you'll always be farther along if you started with a higher velocity. 
I have not yet considered the complications arising from rotational motion of your chair if you push off at an angle. That is left as an exercise to the reader.
If you choose to perform an experiment, please take proper safety precautions and consider wearing a helmet. Also assess what sort of damage you might cause to the floor and what sort of budget you must set aside to repair it (or how to successfully avoid responsibility for doing so). Or, stop playing with your chair and get back to what you're supposed to be doing :-)
