what happens if you leave the null character out when defining a char array?
You get an array containing just the characters you specify.
What is the importance of the null character in this scenario?
In C, it's conventional to represent a string as a null-terminated character array. This convention is sometimes used in C++ to interoperate with C-style interfaces, or to work with string literals (which inherited their specification from C), or because the programmer thinks it's a good idea for some reason. If you're going to do this, then obviously you'll need to terminate all the arrays you want to interpret as strings.
The question seems to be about C++, although you've also tagged it C for some reason. In C++, you usually want to use std::string to manage strings for you. Life is too short for messing around with low-level arrays and pointers.
Might the absence of null character in the example above cause some problems later on?
If you pass a non-terminated array to a function expecting a terminated array, then it will stomp off the end of the array causing undefined behaviour.
Do you recommend always including or excluding the null character when defining char arrays this way?
I recommend understanding what the array is supposed to be used for, and include the terminator if it's supposed to be a C-style string.
std::string.