For extra fun, the proxy may only start injecting things once you've been using it for a while - it can appear safe to a cursory check.

Most do actually, most links on html are changed so that you continue using the proxy, JS is often removed, etc.

You are referring to a web-based proxy.

Changing advertisements, to take credit for views and clicks, would be quite beneficial. This would could go un-noticed by the viewer for a long time too. I would not be surprised if a lot of these free proxy servers do just that.