Now every input contains previous transaction hash and output index. Instead, we could just replace it by public key in compressed form. The meaning would be "take every previous output matching this key as an input in this transaction". It can be especially useful if there are many outputs with small amounts sent to some address.

Now we have r=(k*basePoint).x in signatures and (privKey*basePoint).x in compressed keys. We have y^2=x^3+7 function, so having given x we can calculate y^2 and then we have two possible y values matching this equation. Instead, when we have some y, we can calculate x=cbrt(y^2-7) and there is only one matching x. So the question is: why x value was chosen instead of y? Are there some performance-related issues?

Any non-coinbase transaction has some inputs. They contain previous transaction hashes, so as long as hash function is not broken, they are always unique. But what about coinbase transactions? In this case all inputs are ignored, so it is possible to create two identical coinbase transactions and place them in two different blocks. What happens when someone wants to spend these coins?