You should count the number of the "good" instances and divide by the total number of instances.
For the first question - if the smallest is $x$ and the largest is $y$, then you are writing words of length $n$ using only the symbols $x,x+1,...,y-1,y$ (why?). How many such words do you have?
For the second question - it's equivalent to counting the number of solutions to $X_1+...+X_n=s$ where $1\leq X_i\leq N$, and then dividing by the total number of words of length $n$ over $1,...,N$ (why?). As to finding the number of solutions to this equation: it's the same as the number of solutions to $Y_1+...+Y_n=s-n$ where $0\leq X_i\leq N-1$. This can be done with exclusion-inclusion principle.
The other question, in your second post, is very similar. You can use the same approach, but there each letter/value can appear only once.