DFSA and NFSA intersection problem
Regarding question (1):
As long as $k$ is $\Omega(n^c)$ for some $c &gt; 0$, then the problem is $PSPACE$-complete.  See this paper by Klaus-Jörn Lange and Peter Rossmanith (1992) for some related results.
https://doi.org/10.1007/3-540-55808-X_33
Regarding question (2):
Say that the intersection problem is parameterized by the number of FSA's.  If this parameterized problem is in $FPT$, then $PTIME \neq NLOGSPACE$.
In fact, the result is quite a bit stronger than that.  Essentially, it says that if for every $\alpha &gt; 0$, there exists $k$ sufficiently large such that the $k$th slice of the parameterized problem is solvable in $O(n^{\alpha \cdot k})$ time, then $PTIME \neq NLOGSPACE$.
See here and here.
Regarding question (3):
I am still trying to understand this question.
Regarding last question:
Both the DFA and NFA parameterized intersection problems (when parameterized by the number of automata) are fpt-equivalent.  In fact, the fpt-equivalence is very strong and preserves the parameter (I called it an LBL-equivalence).  Regarding your question, this implies that if one of them is in $FPT$, then the other is in $FPT$ too.
Additional information:
Part of the reason why we have these hardness results for the parameterized FSA intersection problem is because it is $XNL$-complete under a strong form of fpt-reduction.  $XNL$ is a parameterized complexity class that is related to $k \cdot log(n)$ binary space nondeterministic Turing machines.