Dropped tables are not going to Recycle Bin

It's likely that recyclebin isn't turned on. Check in your init.ora file to ensure it is. You can check your parameter:

select *
from   V$PARAMETER
where  NAME = 'recyclebin';

And turn it on at session level...

ALTER SESSION SET RECYCLEBIN=ON;

...and then try a drop.

Also, check both USER_RECYCLEBIN and DBA_RECYCLEBIN for entries.

EDIT: I see you've updated to say you are logged in as the SYSTEM user. For some reason, flashback/recyclebin does NOT work for SYS. I can't find a good reason why but it just doesn't. Create a new user and try creating and dropping a table while logged in as them. I'm sure you'll find that the recyclebin works fine then.

Be sure the ORACLE_HOME and ORACLE_SID environment variables are correctly set. Then run sqlplus / AS SYSDBA as a user who is a member of the oradba group on the computer.

Get the machine down forcibly:

SHUTDOWN ABORT

Start it up, but try not to let stuff run wild:

STARTUP RESTRICT

Shut it down more gracefully. Don't stop until recovery is complete.

SHUTDOWN IMMEDIATE

Start it up.

STARTUP