Create the DB, make the DB files part of the installer, copy them into the right SQL Server directory, hope it recognizes it (will it?)
It will not, you will have to attach the database to your server afterwards. You could add a sql script that will attach the files to your server though.
Let the application test for the existence of the DB, and, if it ...
You do not have to install an Oracle Client for following cases:
Your application is based on Java/JDBC, like SQL Developer
Your application uses the Oracle Data Provider for .NET Managed Driver (ODP.NET Managed Driver)
Your application uses a third-party Oracle Driver, e.g. Devart or Progress (but I am not 100% sure about that)
For almost any other ...
You haven't installed any client because SQL Developer is a Java application, and it uses the Oracle JDBC driver by default, which is located under jbdc/lib inside the directory of SQL Developer and it is included in the zip file you downloaded. SQL Developer is able to use an Oracle client as well.
Yes, you need to have something for connecting to Oracle ...