The important bit is that the compiler must be able to prove that the code has no side effects before it can remove it (or determine which side effects it has and replace it with some equivalent piece of code). In general, and because of the separate compilation model, that means that the compiler is somehow limited as to what library calls have observable behavior and can be eliminated.
As to the deepness of it, it depends on the library implementation. In gcc, the C standard library uses compiler attributes to inform the compiler of potential side effects (or absence of them). For example, strlen is tagged with a pure attribute that allows the compiler to transform this code:
char p[] = "Hi there\n";
for ( int i = 0; i < strlen(p); ++i ) std::cout << p[i];
into
char * p = get_string();
int __length = strlen(p);
for ( int i = 0; i < __length; ++i ) std::cout << p[i];
But without the pure attribute the compiler cannot know whether the function has side effects or not (unless it is inlining it, and gets to see inside the function), and cannot perform the above optimization.
That is, in general, the compiler will not remove code unless it can prove that it has no side effects, i.e. will not affect the outcome of the program. Note that this does not only relate to volatile and io, since any variable change might have observable behavior at a later time.
As to question 3, the compiler will only remove your code if the program behaves exactly as if the code was present (copy elision being an exception), so you should not even care whether the compiler removes it or not. Regarding question 4, the as-if rule stands: If the outcome of the implicit refactor made by the compiler yields the same result, then it is free to perform the change. Consider:
unsigned int fact = 1;
for ( unsigned int i = 1; i < 5; ++i ) fact *= i;
The compiler can freely replace that code with:
unsigned int fact = 120; // I think the math is correct... imagine it is
The loop is gone, but the behavior is the same: each loop interaction does not affect the outcome of the program, and the variable has the correct value at the end of the loop, i.e. if it is later used in some observable operation, the result will be as-if the loop had been executed.
Don’t worry too much on what observable behavior and the as-if rule mean, they basically mean that the compiler must yield the output that you programmed in your code, even if it is free to get to that outcome by a different path.
EDIT
@Konrad raises a really good point regarding the initial example I had with strlen: how can the compiler know that strlen calls can be elided? And the answer is that in the original example it cannot, and thus it could not elide the calls. There is nothing telling the compiler that the pointer returned from the get_string() function does not refer to memory that is being modified elsewhere. I have corrected the example to use a local array.
In the modified example, the array is local, and the compiler can verify that there are no other pointers that refer to the same memory. strlen takes a const pointer and so it promises not to modify the contained memory, and the function is pure so it promises not to modify any other state. The array is not modified inside the loop construct, and gathering all that information the compiler can determine that a single call to strlen suffices. Without the pure specifier, the compiler cannot know whether the result of strlen will differ in different invocations and has to call it.