Generally ... most people fuse everything down, then do the top stitching.
If stitching all with one thread colour, it's not as critical as to which order you stitch.
If multi-thread-colours, then you need to think through what would be on top of each other, IRL.
For your example, for the box, you would stitch it, with thread breaks when you come to the straw.
Then stitch the straw which will in effect, overstitch what your box stitching.
Similarly, your chickens would be stitched before the straw.
Clear as mud?
All that being said ... this is only generalities,
and there are other variations ... or develop your own style!