When do I need to install Oracle database client?
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 databases.
It can be a JDBC driver, a .NET provider, or the collection of libraries that is called the "client".

32 or 64 bit depends on the platform and application you use. For modern tools and application you typically go for 64 bit.