The only instance of a potentially uninitialized local pointer is in LinkedList::deleteNode(double). If nodePtr != NULL but nodePtr->value == num, the loop won't be run, the if statement will run, and previousNode will not have been assigned a value.
Please please please in the future use code tags. You can get them by clicking the little <> button to the right of the text box you make your posts in, and putting your code in between them.
I think TheRabbitologist is right with his explanation of why the compiler thinks the pointer can be NULL, but a slightly deeper look shows that the compiler is wrong. Lines 10-12 handle deleting the first node so line 17 will always execute.
You can shrink all of this code enormously with a trick. Instead of using a pointer to the previous node, use a pointer to the pointer to the node. In other words, a pointer to head, or the previous node's next pointer: