Why can't we assume the positive direction is upwards when solving the DE motion of air drag? In essence my question is: Why is it that in differential equations that motion of the object has to be assumed as the positive direction? is there a way to write DEs with the upwards direction as positive which will lead to the right answer? For instance: Almost all reference material lists: m(dv/dt) = mg - kv but what if I wanted to interpret the positive direction as upwards: m(dv/dt) = kv - mg when solving the Differential eq this way, there is no transient term (terminal velocity cannot be achieved).

You can define the axes whichever direction you want as long as you are consistent. You are correct that the first form has the axis positive downward, which makes $g$ be positive and the velocity positive. If you define the axis as positive upward and take $g$ to be a positive number the equation should be $$m\frac {dv}{dt}=-mg-kv$$ because the change in velocity is opposite to the velocity. You can then compue terminal velocity by setting the derivative to zero, getting $$v=-\frac {mg}k$$ This is negative as it should be because the object is moving downward.