Dynamic resistance of diode
Your misunderstanding is around how the small-signal AC characteristics combine with the large DC characteristics. If VD is the DC (ie: average) voltage across the diode, you've already assumed that to be 0.6V. 

The value rd represents only the incremental resistance of the diode if you add or subtract a small amount of current (ie: add or subtract a small amount of voltage at the input, like 0.1 cos(wt) ). So you use rd to calculate the voltage resulting only from that small change in current.

So:

vin = 0.1 cos(wt), and the resulting small signal effect across the diode is:

$$v_d = \frac{v_{in} r_d}{r_d + 1\mathrm{\ k\Omega}}$$

and Vdiode_total = VD + vd