What are valid publications to put on a grad school application resume?
The type of publications that you should list in the publications section of an academic CV are those writings that contain technical work at the level of a typical scientific publication.  This can include non-peer-reviewed and unpublished works, but they should be the sort of thing that, if circumstances were different, might be reasonable to send through peer-review.

With regards to the examples that you list, then:


Confidential company research paper: this would be reasonable to put in a non-peer-reviewed research section, with a note that it is not available due to confidentiality restrictions.
Company teaching materials: these would go in "teaching experience," not publications.
University research project papers: these would generally not be listed, unless something above and beyond the usual happened, like they were actually sent off for peer reviewed publication.