$n^*=7$.
The third question does not seem to have a pleasant answer.
For instance, there are 2752 completions to 7 entries of 

$\matrix{
*****\ *\cr 
*****\ *\cr  
**6**\ *\cr  
*****\ *\cr 
*****\ *\cr 
*****\ 6}$

that have unique solution.