Welcome to the Atlassian Community!
If you have Scriptrunner, then yes, you can add a "fragment" to the menu that can lead you to run a script or take you to an existing issue action. (Note, answer is for Server/DC only as you flagged it with Server, this can't be done on Cloud yet)
Although, are you thinking about a workflow transition? The action of changing the status of the issue? Those appear in their own area, and are controlled by permissions and conditions. You can edit the workflow to add/remove transitions in the list.
Yes, you can have a rule that triggers on the action, depending on what it is, as long as it fires an event your rule can see and act on.
You must be a registered user to add a comment. If you've already registered, sign in. Otherwise, register and sign in.