Yes but you would change the order in that case for it to sound right. It would be “Why do you have eye bags that big?” Other examples: Why do you eat that much? Why are you running that fast?

Actually we don’t say ‘eye bags’, we say ‘bags under your eyes’ - so a more native way to say your phrase is “why are the bags under your eyes soooo big? You need more sleep dude!” Or “why are those bags under your eyes so massive?” Your answer would be “It’s hereditary”, meaning you inherited this look from your mother or father.

You can, but you need to use 'those' and not 'that' because bags is plural.