How to check if Oracle DB process is running in Unix and where Oracle DB is installed
This depends on the flavor of *nix where you run on and whether the installation was customized or not. If something of Oracle was installed on your system, the directory /etc/oracle or /var/opt/oracle normally would have to exist. Normally they contain a file /etc/oraInst.loc or /var/opt/oracle/oraInst.log that points to the inventory that contains the central registry of the installations done on the host, if it has been done in a standard way.

If you found the oraInstloc you know at least that some installation has been taken place. The inventory contains the details about what was installed and where. Normally this directory is protected.

If the /etc/oratab or /var/opt/oracle/oratab file has been maintained, it contains a list of all instances running on your system, including the software locations. This is the file that is used by the oraenv utility that sets the minimal environment variables you need to be able to use that software for the specified ORACLE_SID.

The oratab, if maintained shows all defined ORACLE_SID's, also when they are not running.

But again, asking your dba might save a lot of time.