1) I have database in same Logical Server. How to move the DB under Elastic Pool?
You can do this in the Portal, Using PowerShell, or as I preferred using simple query
ALTER DATABASE Your_DB_Name MODIFY ( SERVICE_OBJECTIVE = ELASTIC_POOL ( name = Your_Pool_Name) ) ;
2) I also have database in different Logical Server. How to move the DB under Elastic Pool? (Should I restore from back-up?)
Copy the database using simple query (execute it in the new server, and check the documentation for more information)...
CREATE DATABASE Your_New_DB_Name AS COPY OF Server_Name.Your_Original_DB_Name ( SERVICE_OBJECTIVE = 'Basic' );
3) Is it possible to move database in Elastic Pool to outside Elastic Pool in same Logical Server as Standard tier DB?
Yes, exactly like we add to a pool we can remove from the pool using the same ALTER command. Simply select a different type like BASIC for example
ALTER DATABASE Your_New_DB_Name MODIFY (EDITION = 'Standard', MAXSIZE = 250 GB, SERVICE_OBJECTIVE = 'S0');
Same link as above in question 1