After launching the Activity Map plug-in, the Clicks of some links are showing as 0. However, the destination page in a different domain shows up when clicking these links. Does Activity Map work on cross-domain tracking?
No. Activity Map only works on the same domain.
When you click a link on the page, Activity Map stores the link data into a first-party cookie. And then set into Activity Map context data variables which will be sent with Analytics page view calls on the subsequent page.
For the other questions related to Activity Map, please refer to the following document.