Tell and explain why the minimum vertical distance from a to b is 104nm

First of all . . . I agree with your answer of 633 nM for the
wavelength in air.  Good work.

Now . . . (a) (ii) says that light from 'A' is 180° out of phase
with light from 'B'.  

I puzzled over this for a while, but I see it now: 
They're 180° out of phase anywhere below point-B.

In order to reach anywhere below point-B, the light reflected
from 'A' had to travel farther than light reflected from 'B'.
How much farther ?  Up from 'B' to 'A', then down from 'A' to 'B'.

By the time the light from both of them gets reflected and reaches
anywhere below 'B', the light from 'A' has traveled farther than
the light from 'B', by an amount that's double the distance from
'A' to 'B'.
That makes the light from 'A' 180° late.
180° = half of a wavelength.
So the distance from 'A' to 'B' is 1/4 of a wavelength.
The wavelength in the transparent plastic is  414 nM.
1/4 of that is  (414 / 4) =  103.5 nM .

I see under the question were you calculated the half-wavelength. 
You were on the right track.  I did the same thing at first.  Then I took
your hint of "104 nM", noticed that it's a quarter-wave, and then the
light-bulb went on.