Yes you do need to declare everything including half eaten bags of chips. It's rare that NEXUS has been revoked for this kind of thing but it's not unheard of.
No, when returning to Canada, you don't need to declare stuff you already had with you when leaving. If you get stopped you'll have to convince the CBSA officer that your bottled water originally came from Canada. If there is French on the labels, I bet you'll be fine.
Yes you can take school supplies across in the NEXUS lane. Those are not commercial goods. But you must delcare anything you bring to the USA that you plan on leaving there.