Improper color reproduction after digitizer replacement.

the smart cover not working is likely due to the home button being replaced. A lot of the digitizers available that have the home button included don't support the smart cover function, which is unfortunate. The original home buttons can be used in repair but oftentimes does not make sense because the cost of the part with the home button installed is cheaper than the effort required to "dress" a digitizer with the IC and home button. Its also beyond the skill of some the low rent repair shops.

I'm curious about the change in the display. The display should not have been replaced unless it had been damaged in the course of the repair, if the display had not been replaced, you'd see no difference in the color sharpness and accuracy etc.

As for is the parts used likely "fake"? I can almost assure it. Apple does not sell parts for their iphones or ipads to anyone. There are parts that come out of the "back door" of the same factory that makes parts for Apple that are identical, there are parts that come out of factories that are not associated with apple in any way, and there are parts that are scavenged off of devices. If I had my choice of LCD, I'd choose the scavenged part but we can debate all day about where the "best" parts come from. So yes, if your LCD was replaced in the repair, its most likely a copy, sounds by your description to be not a very good copy.