Is it true that if you solve an NP-complete problem in non-polynomial time, the solution also solves other NP-complete problems as well?
If you solve a Problem that is NP-complete the solution to this problem can not be applied instantly to any other Problem that is NP-complete. However, the problem that you solved might be used to find a solution for other problems in NP.
For example:

If you can solve the TSP Problem your an adaption of your solution could also solve the problem of finding a hamiltonian circle in a graph
If you can solve the hamiltonian circle problem an adaption of your solution could also solve the long path problem
...

So your solution can not be directly used to solve other problems.
But if you found a solution for a problem that is NP-complete this implies that there are solutions in P for every problem that was believed to be NP-complete. This is because NP-completeness is defined as follows:

A problem P is NP-complete if it is NP-hard and the problem lies in NP
Beeing NP-hard means that every other problem P' that is NP-complete can be reduced to the problem P (meaning exactly that if you could find a solution for P you could also find oune for P')

The answer to your second question is simply &quot;No&quot;.
The only reason for a cancidate to solve another problem that is NP-complete is that he could not find a solution for the given problem but for the other problem. And in addition he knows how to modify the problem he is abel to solve in such a way that he finds a solution for the given problem.