I know that calloc allocates memory and writes zeroes to each cell, so my question is: is there a difference between using calloc or using malloc and running over the cells writing NULL to them? Are the zeroes of calloc equivalent to NULL?
No, they are not always equivalent, but on most popular machines you'll be fine. calloc writes a bit pattern of all-zeros to the allocated memory, but the null pointer value might not be all-bits-zero on some machines (or even just for some types on some machines).
Check out the Null Pointers section of the C FAQ for lots and lots of information.