Yes.

You are the design creator. You get to decide. You just need to be prudent in how you mix the colours and make sure that you pull those same colours into other areas of your bathroom. Grey, black and white fixtures could go beautifully together as will varying shades of any of the same colour family. As bathrooms are usually smaller rooms, try to use bolder colours at a minimum. But, there is almost always a way to blend different colours together. But, if you are having to use what you have and you just don't think the colours are working together, try putting a barrier, such as a full or half wall between fixtures to ease the colours together.

generally yes, but there are exceptions--for example if toilet and tub are same color you might use a clear or colored glass vessel sink.

It depends largely on the design and feel of the room. Not a hard rule, but since most people use white tubs and toilets, the sink doesn't matter so much, depending on proximity.

I think if the tub & toilet match you can go with a different sink , using a different finish helps it look more planned out. I love a converted dresser with drawers for instance with a vessel sink in glass or recessed sink made of metals.