I agree with David. See if it's recognized in iTunes. If you can hear it charging, and it's recognized, then it's most likely a display problem. Hopefully the lcd cable came loose. If not, you need a new LCD display. Other than that, could be the backlight, but I'd try the LCD first.
I think it's there for the heat, like you said. I've assembled them without it, and it works fine. But, like you said, the tape isn't cheap. They aren't putting it in there for the !&&* of it. I'd try reusing it if you can...
Mini's are very finicky about how they charge. You can't use just any power block. It has to put out enough watts to charge. I've also used cheap aftermarket cords that won't let it charge. Does it say not charging? Because if so, it's not the iPad, It's the charger or cord. Also, just a heads up, computers won't charge it.
I think you already answered your question. You are right that they are to hold down the Smart Cover. The mini works perfectly fine without them. If you can live with the situation it's not going to hurt anything. If she gets a Smart Cover it just won't hold itself together as well as with two. But, just checking, did you check the back of the digitizer for the other magnet? They get stuck on there sometimes. I can't imagine that it fell out unless it's been opened before.
I agree with Ron. iPad 3 dos not have a ribbon cable. I it is an iPad 4, you should be able to press the home button without having the digitizer completely in place. If it's an iPad 3 you can us a piece of metal to bridge the gap between the two pins on the charging port. Just make sure that the digitizer, and lcd screens are connected to the logic board. I've run into problems from energizing the iPad without them in place.
It should work, even detached. It doesn't have the metal pins like the larger iPads to make contact with. Just be careful not to press it too hard, or it will damage it. Or, you can hold the back of the button for support and press the button.
It is possible depending on how damaged it gets. If it's cracked enough, be very careful of any water or rain. And I highly recommend using a decent replacement assembly. Installed a cheap one a customer brought in and the LCD is horrible. It's very dark, and can only be viewed from a straight on angle.