Can we force a reversible reaction in equilibrium to achieve 100% completion?
The two questions are sort of related, so i'll answer to the best of my ability.

A reaction in dynamic equilibrium can get close to completion depending on its equilibrium constant, which is a ratio between products and reactants at a certain temperature. No, the equilibrium constant won't change if you add more reactants in an attempt to push the reaction to completion. We refer to Le Chatelier's principle which states that if one were to add another parameter, say a reactant, the reaction will act to stabilize the stress that has been placed on it. 

Take this reaction $\ce{PCl3 + Cl2 &lt;=&gt; PCl5}$. If I were to add more $\ce{PCl3}$, the change we would witness is a shift in equilibrium, that is to say, the system will react to consume more reactants and produce more products. What we see essentially is the system reacting to stabilize the changes we made. 

This principle isn't just limited to adding reactants or products, however. We see the system tries to stabilize changes made to pressure, temperature, and volume. If we increase pressure, how will the system act to stabilize it? Pressure is a result of the gas particles hitting the walls of the container. Thus we would see the system reduced the number of gas particles in order to reach equilibrium again. Essentially, it will shift to the side with the least amount of gas particles. Volume and pressure are obviously related in this principle. An increase in volume is a decrease in pressure and a decrease in volume is an increase in pressure. 

Take this reaction for example $\ce{CaCO3(s)&lt;=&gt;CaO(s) + CO2(g)}$. If we increase the volume (decrease the pressure), we would see the system shift to the right because it has the most moles of gas and we need more gas particles to fill the volume and stabilize the pressure.

Lastly, when changing heat we view it as a product or a reactant for exothermic and endothermic reactions respectively. An exothermic reaction has heat written as a product $\ce{A &lt;=&gt; B} + \text{heat}$ and an endothermic reaction has heat written as reactant. Thus, when we change the heat of an endothermic or exothermic reaction, we are changing its reactants or products and we treat it in the same way we treat a change in concentration.

The point is that the equilibrium constant remains constant, otherwise it would be called the equilibrium variable. We do, however, see a change in equilibrium position.