Given any positive integer $n$, do there exist functors (endofunctors if $n$ is odd) $F_1, F_2, ..., F_n$ for which $F_1 \dashv F_2 \dashv ... \dashv F_n \dashv F_1$, but none of the $F_i$s (with indices of the same parity unless the involved functors are endofunctors) are naturally isomorphic to each other?
For $n=1$, the identity functor on any category is left adjoint to itself. For nontrivial examples of self-adjoint endofunctors, the functor $A \mapsto A \oplus A$ on the category of abelian groups (or any additive category) would work.
For $n=2$, one would get ambidextrous adjunctions. A well-known example of an ambidextrous adjunction is the direct sum functor $(A, B) \mapsto A \oplus B$ being simultaneously left and right adjoint to the diagonal functor on the category of abelian groups (or any additive category).
For $n=3$, I could not think of any category $C$ with three mutually non-isomorphic endofunctors $F_1, F_2, F_3:C \to C$ for which $F_1 \dashv F_2 \dashv F_3 \dashv F_1$.
Yes, there exist $n$-periodic chains of adjunctions for every natural number $n$. There also exist non-periodic chains of adjunctions of arbitrary/infinite length. This is the topic of Booth's 1972 paper Sequences of adjoint functors.