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 $908\,928$.
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 $396\,800$ valid clues with core size 2 and they form $6!$ times $12\,242$ $G$-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 $496\,128$ valid clues with core size 4 and they form $6!/2$ times $5320$ orbits under a group which is twice as large as $G$.

$396800 + 16000 + 496128 = 908928$.

The programs we wrote for this task can be found at 

https://github.com/wilberdk/sixy