Professional edition can have up to 5 process builders, but their immediate action doesn't contain "Apex". If we install a managed package which has a global invocable method, will the Apex appear as an immediate action in process builder or not? Does anyone know about this or have exposed the invocable method for the Professional edition through a managed package.
No, Apex will not appear among Process Builder's actions in Professional edition.
It is noted in this question: Is Process Builder available in Professional Edition?
To sum up, you won't be able to use your invocable method in professional org.