What happens to interest rates when they are already at (near) zero and the stock market crashes?
They would be just pushed to their lowest possible level. Recent experience in Switzerland and Germany shows the interest rate can be slightly negative but that’s it. Once the interest rate hit (zero) lower bound it can’t be pushed further as long as there is cash that carries implied zero interest. At that point central bank would have to resort to other policy instruments such as QE, helicopter money, forward guidance etc.

However, if some western country would decide to go completely cashless it would be possible to have arbitrary negative interest rate. In that case the interest rate would just get pushed further down.