I used to be able to copy a shape in Illustrator, paste it in Photoshop, and then duplicate the shape layer, removing what I wanted from each layer. Now that doesn't seem to work, and what's worse, strokes are all messed up - if I apply a stroke to the whole layer, it isn't all visible. If I delete parts of the layer that are unwanted, certain pieces remain invisible, even if I apply a stroke (and fill, for that matter).
You could try to save the file as pdf if you want to take the whole content as one from AI and open the same in Photoshop.
Or save the file as Photoshop and while making the files make sure you put the content in different layers while creating in Ai and then save as PSD (Photoshop Document)
While save make sure you have write layers option selected when you save as PSD in advance options