I'm no expert, and haven't even done anything like this personally, but if I was trying to tackle this I would draw the grid going from top to bottom, left to right. So at location 1,1, draw everything in that location based on a layer number, with arrows, birds etc having a high layer number so they are drawn last, then move onto the next grid and repeat.
As for locations / images, rather than the arrow image being 5 pixels high and 30 pixels wide - to simulate it flying across the screen, I would make the image something like 100 pixels high and 30 wide, and draw it at the grid location minus the height - that way it will draw over the grids above.
Hopefully this will give you an idea, but I'm sure some of the more experienced game developers here will give you better tips. I'm interesting to know if I'm on the right path with the above too.