Note that the third side must be less than the sum of the other two, so it must be less than $23$. But the third side is an integer, so its largest possible value is $22$. It is easy to see that $22$ is achievable. Just put a hinge where the sides $9$ and $14$ meet, and open up the hinge so that the sides $9$ and $14$ almost form a straight line (that is, make an angle that is almost $180^\circ$. If you wish, you could compute the suitable angle by using the Cosine Law.
As to the smallest value of the third side, we need to make sure that whatever it is, it plus $9$ is bigger than $14$. The smallest integer that works is $6$.
The difference between the largest possible perimeter and the smallest possible perimeter is therefore $22-6$, since the other two sides are the same for each triangle.
If $h$ must be an integer, then the perimeter must be an integer, and the answer of $17.5$ is not possible.
Remark: The Triangle Inequality (the sum of any two sides must be greater than the third side) is an important fact about distances.