including 一 feels clumsy, is it required? I generally use 下个 for next and 上个 for previous.
No, its actually colloquial to use 下个项目 instead of 下一个项目 since as a 量词, it should be 一个项目.
I don't think it's required, but it's also not out of the ordinary, and may even sound more natural in some contexts.
What about "event" (as in "sporting event")? Would that require a clarifying modifier, or is it a valid translation of 项目?
It is the first thing I think of so I put that, but was marked wrong.