I have to create a new database for a company wide printing
application.Since there is not much overhead, we are planning to use
an existing SQL Setup for the same. Can someone suggest when can I
actually move the db to the AAG group once created?
Yes, You can add the database in AAG immediately after creating the database. But there is condition, First you have to take the 'Full Backup' of that database(which database you want to add in AAG, and that Database Recovery model should be 'Full Recovery model' . otherwise that database will not available to add in AAG, untill unless whenever you shall not take full backup.
Also, for configuring the secondary database, if full initial
synchronization is enabled, will it be created automatically?
If your have configure properly 'primary & secondary replicas' in 'availability groups' . And if you shall add your 'Primary Replica Server' database in 'Availability Databases' . It's automatically comes that database in your 'Secondary Replica Server'. After successfully adding the database both server (Primary Replica Server & Secondary Replica Server ) the status of that database would be "Synchronous commit or Asynchronous commit". Whatever you shall choose during 'Availability Mode' configuration in your Availability Groups.
Hope it will help out to you.
For Ref :- Here