Why do functional tests require a running website?
You can theoretically run a functional test on a production website. (I am not saying you should though.)
In my experience the testing framework will install a new Drupal site for every individual test and in the end remove these sites. This is done by creating testing tables in your existing database, for which you need to provide the credentials in SIMPLETEST_DB.
As long as the tests do not error, all of this data will be cleaned up at the end of the test and you do not need to worry about anything.
If the test unexpectedly stops you might need to clean op some items manually (previously could be done with simepletest module, but I believe there are plans to remove that).
I have no idea if you can use a different site. You need to have database access to that site to install the testing tables. Modules should be available, etc. I do not think it would be reliable.