Can I host an exchange student if my family divorced?

The family of the foreign student would have to know about it and agree.

Ask at your school for advice. And check with both your parents that they can host the student.

I ran the idea by our current exchange student who has been with us for two weeks now, and he was not sure that he would be able to cope with the changes. It's weird enough adjusting to one house, and one set of house rules, in a foreign language.

The way to find out is to apply to be a host family. Everyone in both households would need to agree, and if I were a representative of the exchange program I would also disclose this arrangement to the family of the prospective exchange student. It is NOT likely to work unless the student and his family are not comfortable with the idea.

Alternatively, could you rearrange the custody schedule to have at least one semester (preferably second semester if this will only work for a semester) in one place, perhaps with weekends with the other parent? Single semester students do better if they come in at 2nd semester and ride out the end of the school year with their peers. The local students are already settled in, any full-year exchange students already have made American friends, so it is a lot easier for 2nd semester students to make friends quickly.

Yes!

There are still students that need a family for this school year.

Contact a local volunteer as soon as possible!

I'm sure it would be fine, just look at more than one hosting program.