What IP passes along?

yes it does. But if you used a referer cloaking then
they wont know what page the person came from.

but when the person that clicked the cloaked link, end up on the destination page, will the IP they're using be displayed for the network to see?

Yes, anything happening on the landing page of the network is visible for the network.

IP, browser, screen, language, etc.

A simple on-site cloak only hides the affiliate link from your visitor.
So instead of yourdomain.com/IEarnMoneyFromYou673824
they see something like yourdomain.com/recommends

The referrer, that is yourdomain.com, is passed along to the networks site.