How does a database use the SYS schema even when we change the password

A trigger on schema (with no schema specified) doesn't fire for all schemas. It only fires when the triggering action is run by the user who owns that trigger.

So they are quite different, and are not interchangeable.

Here's an example that sets up a logging table, and three create triggers: on database, and on schema for users foo and bar:

Connected. -- as mat
SQL> create table mat.log (dt timestamp, who varchar(3),
  2      cur varchar(10), own varchar(42), obj varchar(42));
Table created.

SQL> create or replace trigger db_trig
  2  after create on database
  3  begin
  4    insert into mat.log values (systimestamp, 'db', user,
  5      ORA_DICT_OBJ_OWNER, ORA_DICT_OBJ_NAME);
  6  end;
  7  /
Trigger created.

SQL> connect foo/foo
Connected.
SQL> create or replace trigger foo_trig
  2  after create on schema
  3  begin
  4    insert into mat.log values (systimestamp, 'foo', user,
  5      ORA_DICT_OBJ_OWNER, ORA_DICT_OBJ_NAME);
  6  end;
  7  /
Trigger created.

SQL> connect bar/bar
Connected.
SQL> create or replace trigger bar_trig
  2  after create on schema
  3  begin
  4    insert into mat.log values (systimestamp, 'bar', user,
  5      ORA_DICT_OBJ_OWNER, ORA_DICT_OBJ_NAME);
  6  end;
  7  /

Trigger created.

Now let's create a table in foo's schema, as foo:

SQL> connect foo/foo
Connected.
SQL> create table foo.foo_stuff (id number);
Table created.

And let's create a table in foo's schema, as bar:

SQL> connect bar/bar
Connected.
SQL> create table foo.bar_stuff (id number);
Table created.

Here's what we've logged:

SQL> select * from mat.log order by dt;

DT                             WHO CUR        OWN        OBJ
------------------------------ --- ---------- ---------- ---------------
25-NOV-12 07.52.03.797794 PM   db  FOO        FOO    FOO_TRIG
25-NOV-12 07.52.03.828670 PM   db  BAR        BAR    BAR_TRIG
25-NOV-12 07.52.03.865334 PM   foo FOO        FOO    FOO_STUFF
25-NOV-12 07.52.03.865579 PM   db  FOO        FOO    FOO_STUFF
25-NOV-12 07.52.03.894672 PM   bar BAR        FOO    BAR_STUFF
25-NOV-12 07.52.03.894911 PM   db  BAR        FOO    BAR_STUFF

6 rows selected.

So:

• the two create trigger statements were logged by the "global" after create on database trigger. That trigger also logged everything else.
• foo's after create on schema logged the table creation that was done by foo
• bar's trigger logged the table creation that was run by bar himself, even though bar created a table in foo's schema.

You can do this with the SQL*Plus password command.

To change the current user's password, just type password and you'll get the usual prompt.

$ sqlplus foo/foo123
...
SQL> password
Changing password for FOO
Old password:
New password:
Retype new password:
Password changed

You can also use password username to change another user's password (if you're logged on with sufficient privileges - you need to be dba I believe).

SQL Developer also allows you to do that. Just type password and execute, and you'll get a password change popup. Or expand the "Other users" branch, right-click on the account that needs a change and select the "Edit user" menu.