Why is this chair conformation the most stable and which leaving group is best? I'm confused about how this chair conformation was constructed and why that particular Cl leaves. Why is one of the Cl's drawn axial while the others are drawn equatorial (it seems like the others could have been drawn axial as well)? And, to reiterate, why is it that Cl that's favored to leave over the other two?
In cyclohexane, the equatorial position is energetically favored over the axial position. So the lowest energy conformer is the one where the most substituents are in equatorial position. (Or rather: Where you minimize the energy according to the A Value).

The axial Cl is favored as leaving group because of the elimination reaction mechanism. Normally, we see $\beta_{1,2}$ elimination in an anti-fashion. Which is what is proposed by your image.