Can we connect this apex to other Databases installed on Different
Machine.
Depends on what you mean with this. If you just want to get (additional) data from another DB then you can do this with a database link (Docs: administrator's guide: database links).
Can we install Apex without installing Oracle DB in the system for
instance only installing apex in the system and later connecting it to
DB in other system?
No. Apex is not some standalone executable, it is a database product. You can not have apex without the Oracle DB.
Apex Docs: 2.1 Oracle Database Requirement
Oracle Application Express release 4.2 requires an Oracle database
(Enterprise Edition, Standard Edition or Standard Edition One) that is
Release 10.2.0.4 or higher. Oracle Application Express 4.2 can also be
used with Oracle Database Express Edition 10g Release 2 and later. The
Oracle HTTP Server is not included with the distribution of Oracle
Database Express Edition.
Take a good look at the requirements so you know what you're getting in to. For example: what listener will you use?