How to start running a project in parallel on several machines with appriopriate delays?

It depends a bit on what you're doing with your test data as to whether or not it is really a concern.  However, in short, look up the topic on "Synchronization Points" (https://support.smartbear.com/testcomplete/docs/testing-with/advanced/distributed/synchronizing/index.html) to get information about how to get Slave projects to interact and sync up.