What version of the product are you using? I understand with v11.52 there is a feature within the ALM UI where you can schedule tests to run at a given time. In earlier versions of the product, through the UI you can't set up a schedule as a background task. When you use the Run Test Set button in the Test Lab, the tests run now. If you add an execution condition that a certain time has to have passed, you still have to click Run Test Set and then leave the UI in that state until the time passes.
Through the UI you should also be able to set up a Host Group that includes you 5 QTP systems. Then specify the tests to run on the host group as their Planned Host.
You can use the OTA API to access the Scheduler object and start test execution that way. Then you can use another tool, like Microsoft Task Scheduler, to set your program up to run at a particular time. If you've written you're own program to do that, then you can also put code in there to read a file of host names. There is sample code in the OTA API documentation that comes with the product. There is also a knowledge base article on the HP Support web site, but you have to have a support contract to access that. In the KB they actually provide a copy a a program that uses the OTA API to run tests, plus the source code so you can modify it as you see fit.