I think the simple way to find the target database running on the target server , which shows the mount points related to your delphix engine server...
you can see the df-h | grep your delphix mount point/or db name....
No, Delphix won't install anything on source/or target only you need to copy the toolkit and check the prerequisites.. Like delphix user, read/write permission to mount point, sudo privilege to delphix user to mount, unmount the file system, read permission to /etc/oratab... etc.
Toolkit , is some of the delphix written shell /rman scripts.. which do the rman level0/ level1 backup scripts... while you are configuring the Delphix ---dSource which have the details of Source server name, username, password/ dbname etc.. with that it connects from Delphix engine to SourceSever/dbname to get the db changes ( level 0 initially one time) after that it take level 1 or incremental backup/or archivelogs backup to delphix engine..
Target also similar.. you need to configure in delphix engine where your vDB to be created -- like server name , osuser. dbname , mount point etc.
When you are provisioning a vDB all the above details are required ..