Ah we really should have something on our site that explains this - no, there's no dependency on the paid Cypress dashboard. A few teams use us alongside them, most others don't. We've recently added load balancing for parallelism, as well as some basic analytics based on user feedback to help fill in those gaps if you're using just us.
As for Github - no as well. We just have a Github app integration if you do to comment in on PRs. But we have teams using Gitlab/Gitlab CI as well. We print out our debugger links directly in your CI stdout or also add links into your JUnit report if that's enabled (ex for Jenkins). Though there's plenty of teams that just check our dashboard directly for test results as well! We just want to make it easy to access your test failure when they occur.