> Why does it work with sqrt(-1), and not with dividing by 0

This is pretty straightforward to explain.

With dividing by zero, any number you multiply by zero becomes zero. There's no other way to get zero through multiplication. So, the only sensible thing to divide by zero is zero (since "What times zero equals six?" doesn't have an answer).

But because anything multiplied by zero is zero, even if you divide zero by zero you have an undefined answer. Zero divided by zero could be anything! So that's also not meaningful.

The square root of negative one, however, only has to be defined once to get meaning out of all negative square roots.