The time complexity for particular runs of quicksort can be anywhere between the best case and the worse case. It can be $\Theta(n(\log n)^2)$, $\Theta(n^{\frac32})$, $\Theta(n^{\frac{19}{10}})$ or $\Theta(n^2/\log n)$. Given any $f$ such that $n\log n\le f(n)\le n^2$, we can construct runs of quicksort whose time-complexity is $\Theta(f(n))$.
Let $q$ be a quicksort algorithm. Let $c_q(A)$ be the number of comparisons used by $q$ on input array $A$. Let $m_{q,n}=\min_{\#A=n}c_q(A)$ and $M_{q,n}=\max_{\#A=n}c_q(A)$ for $n>0$. The following property holds for various versions of quicksort.
Continuum of comparisons by quicksort: Given any integer $i$ between $m_{q,n}$ and $M_{q,n}$, there is an array $A$ of size $n$ such that $c_q(A)=i$.
We can replace "comparisons" above by "swaps".
Statement 1: Any other partition should result in average case.
Q1. Is statement 1 true? If no, which other pivot selection strategies could result in worst case or best case behavior?
The "continuum of comparisons by quicksort" shows statement 1 does not make much sense. To determine the asymptotic behavior of a quicksort algorithm, we need to specify the version of quicksort with its partition scheme and which kind of input arrays might be given. It does not make much sense to say $O(n\log n)$ is the time-complexity of the average case without specifying "the average case".
Instead of statement 1, here is the clearer summary, where the average case is when the given input arrays are uniformly random, as described here.
All quicksort algorithms (that I have seen, including these variants) takes $\Theta(n \log n)$ time in expectation in the average case.
Some quicksort algorithms choose the pivot that partition an array within some fixed proportion. Those algorithms run in $\Theta(n\log n)$-time in all cases, assuming a linear-time algorithm is used to choose the pivot. However, the constant factors hidden in the big $\Theta$-notations for these algorithms are much larger than those factors for other algorithms.
Statement 2: Any pivot which does not partition array in some proportion should result in worst case. If nth smallest or largest element is selected as pivot, it will result in worst case.
Q2. Is statement 2 true?
Although sounds reasonable, statement 2 is too ambiguous to be verified or refuted. The following is a version that could be proved.
Statement 3: The classic quicksort with Lomuto partition scheme or
Hoare partition scheme runs in $\Theta(n^2)$-time if the pivoting on every array partition the array into two parts with one part of size $O(1)$. In particular, for any constant $c$, if $c$-th smallest or $c$-th largest element is selected as pivot whenever the size is no less than $c$, it runs in $\Theta(n^2)$ time.
Exercise 1. Prove statement 3.
Exercise 2. Prove "continuum of comparisons by quicksort" for a version of quicksort known to you.