"But how is this supposed to refer to a state of mind? Dukkha is not a state of mind. It's a characteristic of the world and of all life."
In the widest sense of dukkha - yes. But the word is often used in a more
restricted sense. In the Four Noble Truths, for example, if I apply them
to myself, then it is my dukkha which has its origin in my craving. It is
just my dukkha which will cease when my craving ceases.
But, the dukkha which ceases is only mental dukkha, not dukkha such as
physical pain - even Arahants still feel that. The dukkha that ceases is
the mental dukkha which arises from ignorance.
The point about the next world is more difficult, but here is an outline.
This world means the three realms ( kama, rupa, arupa). There is a stage
on the path called a non-returner - he is spontaneously "born" into the
pure abodes. These are outside of samsara (cyclic existence) and are the
other world referred to. The Arahant is beyond even this "other world".