We have a customer requirement to activate the child page on rollout from language masters. Currently, author has to first rollout to child geo and then again go to child page and then publish. Instead, has anyone done a custom rollout that would activate the page that was rolled out? Ideally, on the rollout screen, we would like another option to just rollout that reads "rollout and publish" and aem directly publishes the rolled out page
You can try something like below:
Select Language Master with "Manage Publication" option -> "Publish" -> "+ Include Children" -> Select the respective option -> All the pages will shown in the list -> Next and Activate it.
Please go through the blog https://helpx.adobe.com/in/experience-manager/6-3/sites/developing/using/extending-msm.html to get better understanding of creating custom rollout config, You can try creating custom rollout config as suggested by @arunpatidar and add targetActivate as sync action which should activate the rollout pages.
Otherwise you need to achieve your requirement by developing a LiveActionFactory and use it in a rollout configuration.The example is mentioned in the above link.