My plan is to allow folks to submit products, as mentioned. but have the admin approve them before publishing. So they can try to weed out the spam, as graham mentioned, should any be submitted. Is this doable?
For a public submission with admin approval just leave the out the State element from the submission layout. ZOO submission have to be approved by admins per default. They'll get an email notification with a link to the backend to approve the submitted item. The Access Level for the submission can be set to Public in the Edit-Submission-Panel of the ZOO administration.
My other related question was whether or not they would be able to submit without being a member or having to sign up first.
Access Level Public means you don't have to register. See for example my test submission form. Anyone can submit, but I have to approve it first. Note that public submission will always be displayed as submitted by Guest and can never use the Trusted Mode for text areas (that is, they can't submit unfiltered HTML).