Why does a Hashing Function produce good Random Numbers if Input is random?
I mostly agree with your statements.
To answer your questions:


Obviously the entropy reduces if the entropy of $s$ is greater than $2^n$. Using a good hash will however result in that maximum amount of entropy.
Indeed, SHA-1 is not a random oracle. It is a common assumption that a hash function acts like one. Since collisions for SHA-1 have now been practically found that assumption is clearly wrong. The practically used primitives, like compression functions and block ciphers, are thoroughly cryptanalyzed but not proven secure.