As for the first question, a backtracking algorithm, see sixy.c at
https://github.com/wilberdk/sixy
shows there are 1936 completions of
$$\matrix{1&2&3&4&5&6\cr *&*&*&*&*& *\cr *&*&*&*&*&*\cr *&*&*&*&*&*\cr *&*&*&*&*&*\cr *&*&*&*&*&*&}$$
The answer to the second question is $n^*=7$. The answer to the third question is $19\,738\,496$. These are more tricky. For instance, there are 2752 ways to chose five cells in
$$\matrix{ *&*&*&*&*& *\cr *&*&*&*&*& *\cr *&*&6&*&*&*\cr *&*&*&*&*&*\cr *&*&*&*&6&*\cr *&*&*&*&*&*}$$ to construct an initial clue like $$\matrix{ *&*&*&*&*& *\cr *&*&*&*&*& 1\cr *&*&6&*&2&*\cr *&3&*&*&*&*\cr *&4&*&*&6&5\cr *&*&*&*&*&*}$$ that ensures a unique solution. Here the construction fills the chosen cells with 1 through 5 consistent with the lexicographic order.
One sees that $n^*=7$ by replacing $$\matrix{ *&*&*&*&*& *\cr *&*&*&*&*& *\cr *&*&6&*&*&*\cr *&*&*&*&*&*\cr *&*&*&*&6&*\cr *&*&*&*&*&*}$$ with starting configurations that have just one filled cell. There is a symmetry group $G$ of order 128 with just three orbits of cells, and we tried all 6 times 3 orbits of configurations that have just one filled cell. The group $G$ is generated by the following three operations: turn the grid upside down; reflect the grid in the main diagonal; interchange the first two rows.
To be more specific about the answer to the third question, let us make some definitions. Call a clue valid, if it has seven filled cells and a unique solution. Given a valid clue, its core is obtained by deleting all integers that occur only once. The core size of a valid clue is the number of filled cells in its core. For instance the core size of $$\matrix{ *&*&*&*&*& *\cr *&*&*&*&*& 1\cr *&*&6&*&2&*\cr *&3&*&*&*&*\cr *&4&*&*&6&5\cr *&*&*&*&*&*}$$ is two. Possible core sizes are two, three and four.
One tries all $G$-orbits of possible cores.
There are $6!$ times $16\,480\,256$ valid clues with core size 2 and they form $6!$ times $396\,800$ $G$-orbits.
There are $6!$ times $16\,000$ valid clues with core size 3 and they form $6!$ times $226$ $G$-orbits.
There are $6!$ times $3\,242\,240$ valid clues with core size 4 and they form $6!$ times $32\,088$ $G$-orbits.
$16480256 + 16000 + 3242240=19738496$.
The programs we wrote for this task can be found at