Possible to delete a backup db file while it is being restored? I currently a have a db backup, which is very big. So big that I can't restore my db on the same drive.

Is it possible to delete the backup while it is being restored? Such that the new db takes up the space which the backup file was taking?

My backup is currently on our prod server, and there is a link between our test and prod environments.  Could I somehow expose that? Is it possible to restore my db over the net?
Yes, you can restore over the net. RESTORE DATABASE .. FROM FILE = '\server\share...'. I.e., an UNC path. Note that the service account for the SQL Server service need permissions on that share.