Ok from scanning posts here ive come to the conclusion your missing one very fundamental piece of the puzzle regarding basic graphic computation facts.
Because every single post revolve around the same question, your trapped in a zone where you cant see the answer because its in fact invisible. Its fairly common to see people who actually just went ahead and did graphics to arrive here. Unfortunately you must step back a few steps and relearn what you know to be a fact.
Ok your problem regards alpha, and mainly how its shown, funnily while for example photo shop is industry standard does not mean its acting like industry standard stuff.
Since early conception of graphics theres been 2 main ways to understand alpha. It regards how you show alpha to the user, the simplest and lightest way to do this is not to take action untill you must and just show the rgb channels, here usually the background is just set black. However the alphas still there usually applications like this have a button taht enables you to see just alpha channel. Because moving pictures deal with MASSIVE amounts of individual pictures, and also a slightly more pedantic view when it comes to masks that traditional 2d painting applications, mainly because alpha handling is more important to movie artists. To minimize effort the color has been premultiplied into the rgb channels so the image looks good even without associated alpha. (layering on top is too much lighter all you need is subtract the alpha value of the overlaying layer form the underlaying layers color values and add the premultiplied image on top): now the trick is that since the shadow is black you cant see it on the plate, because its black on top of black, however if you switch to your alpha chanel you will see he shadow is white meaning there is a shadow. This ones called premultiplied alpha. And in general unless you can think ahead conceptually in terms of channels you cant see the effect untill you actually layer it on top of something.
Then theres the show transparent as transparent approach, it is usally related to another way to think of alpha namely straight alpha but can actually work on both ways to interpret it. Here the problem is that what does transparent look like? well it does not, so photoshop made a idea that lets layer a abstract grid behind the surface. Well this means more computation, so just showing the image uses up time more, some say its more intuitive. Maybe, but on the other hand it theres considerable amount of time for people to capitalize on the power of using channels properly in this work flow because they never see them (just because you dont know its there don't mean you couldn't benefit form it). Now anyway theres no guarentee you can see the imperfections untill you layer something like a black plate behind something the channel oriented crowd of premutiplied thinking were doing all along.
Whichever is for you is another thing and neither is better (its a optimizations not a better versus worse thing some things are better others arent). But you do need to understand both ways of thinking.
One interesting difference is that a pure photoshop user needs to be taought how to do a key but a person who operates channels thinking images eventually stumbles across the idea of pulling keys out of channels. And youd be supprised hw much more time video artists have spent on trying to make selection automatic mainly because they dont have the time to tweak each picture.
NOTE if you turn primary visibility off you dont get shadows on the floor because its nolonger there to shade.