Computer hardware is agnostic to whatever you put in it, as long as it works with the interface. So no, there's no benefit to sticking an NVIDIA card in an Intel board or an AMD card in an AMD board or any loss going the other way. And no, you do not need an ASUS motherboard to work with an ASUS graphics card.
That is true. There is no problem with mixing and matching video cards and mobos. In Windows, you will have almost no problems.
But, if you play around with linux for things other than basic web surfing, etc., then you have to consider the drivers. With linux, AMD drivers can be more flaky than Nvidia but I hear they are fixing the issues.