Question about lower bound and upper bound key collision
Remember that the bounds are defined for every key. This means, there will be a first set of keys that contains at least $\kappa'$ and at most $\kappa$ keys that all map $x$ to the same $y$. Now, every key not in this set has to be in another set with the same size bounds that maps within which every key maps $x$ to the same $y$ which is different from all the $y_i$ the other sets map to. And so on. 

Hence, there are at least $\frac{2^n}{\kappa}$ such sets of keys. All keys in such a set map $x$ to the same $y$ which is different form the $y_i$'s the other sets map to. Consequently, there are at least $\frac{2^n}{\kappa}$ different $y_i$.

Similarly, there are at most $\frac{2^n}{\kappa'}$ such sets of keys and accordingly no more than $\frac{2^n}{\kappa'}$ different values $y_i$.