$n^*=7$.
The third question does not seem to have a pleasant answer.
For instance, there are 2752 ways to add the numbers 1 through 5 to

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

to get a pattern with 7 keys that has unique solution.