Should you go down a size in climbing shoes?
It depends on the type of shoes and their purpose. Synthetic climbing shoes may stretch up to half a size, and even that is not very common. Usually, there is no noticeable break in difference with synthetic shoes, which means that if you go down a size in them, and they are very tight and painful, that may be how they will always be.
Leather shoes, on the other hand, do tend to stretch, and can even stretch up to 2 whole sizes, some even say more. Because of this, many climbers go down even 2 sizes.