adobe photoshop - Solving problems with a given illustration, how do I proceed?

I've been given this sample image, supposedly "pop art" style. What I'm asked are four steps:

1. reproduce this image
2. control the colors

3. add a two colors logo onto it
4. improve the overall image

1. Reproduce this image

I'm at a good point and I used Filter Gallery > Sketch > Halftone Pattern, the result is not yet OK but quite close to it, I have set the level to Linear Burn and added a yellow background below it, I took the exact color from the image. Edit: Added a Sharpen filter it looks like below, still missing something but probably acceptable as "similar" image to the sample provided:

2. Control the Colors (Stuck Here)

Aside than changing the below background layer I have found no means to alter the black "dots" added by the filter, probably there is another way of adding the pattern which allow more control?

3. Add a Logo

While this is simple, I just need to add two shapes and change the color with some that goes well with this yellow, I also need to reproduce various version of the background/square colors accordingly with the two colors of the logo, this step is the one that worries me the least.

4. Improve the image

I think it is too early to consider this aspect since Im stuck at points 1 and 2 to be honest, the image I get is "less polished" than the one I was given, a possibility to improve could be to add some blurr I think but it is too early to make speculations.

What I wanted to ask is if I'm on the right path and possibly a hint for point 2 where I'm going totally blind, I guess point 2 is inherited from solving point 1, so if my method is wrong I wont make much progress. I need to do this using Photoshop, I think just to make things more complicated.