Springs would also eat some of the power you throw at the impact. Same reason you don't do shocks on a road bike.
(Above is speculation.)
The plate is the full length of the shoe. Not sure how you're imagining adding springs in a way that is more conducive to running. Keep in mind shoe weight matters quite a bit.
Springs are not always shaped like a car strut... AND Weight isn't important if the energy gained offsets the energy lost carrying the extra load.
Sure. I think Nike is probably aware of this and took it into account when designing this shoe. If you have a better design (your earlier comment suggests "just add[ing] springs to the entire shoe"), there's probably a financial motive for you to figure out how to sell that. (I think you are probably understating the difficulty of designing a better shoe, for any definition of "better.")