My projects and activities keep disappearing whenever I change their status to closed. I don't want my tasks listed as incomplete. Is there a way around this?

General Questions
There are two items on the My Options screen that deal with this exact issue. To be able to see completed activities and projects, make sure that these items are both selected.

"Show completed projects"
"Show completed activities"

These options are user-specific, not global. That means that if one user (even an administrator) changes them, the change will not affect what other users see.

Note that a quicker way to enable showing completed objects is to use links in the Quick Panel: