How to check which node should be posted on which domain?
Consider using the Domain Access module. Here's an excerpt from its project page:

The Domain Access project is a suite of modules that provide tools for running a group of affiliated sites from one Drupal installation and a single shared database.
The module uses Drupal's Node Access system to determine what content is available on each site in the network. Unlike other multi-domain modules for Drupal, the Domain Access module determines user access based on the active domain that the user is viewing, rather than which group or site the user belongs to.