Yes. tvOS has no proxy setting, so ProxyBeam uses a configuration profile that sets a device-wide (Global HTTP Proxy) proxy — and Apple only accepts that payload on a supervised device. Supervise the Apple TV once with Apple Configurator (free on the Mac App Store) or Automated Device Enrollment, then install profiles as usual. Heads-up: supervising an Apple TV with Apple Configurator erases it first, so supervise before you set the device up. Step-by-step: how to supervise an Apple TV →
While the profile is installed, the Apple TV's HTTP and HTTPS traffic goes through the proxy. If the proxy app isn't running on your Mac, most apps lose connectivity. Start the proxy — or delete the profile in Settings → General → VPN & Device Management.
Your Mac's IP address most likely changed. ProxyBeam detects the new address automatically — generate the profile again and reinstall it on the device.
No. Delete the old profile before installing a new one — conflicting profiles can break networking on the device.
ProxyBeam is built for Apple TV, where a profile is the only way to set a proxy. The same profile does install on iPhone and iPad, but there it's usually simpler to set the proxy by hand in the Wi-Fi settings — and the profile still needs the device to be supervised, while a manual Wi-Fi proxy does not. So reach for ProxyBeam on iPhone or iPad only if you specifically want a profile.
Any proxy that can export its root certificate as a .cer file — Charles, Proxyman, mitmproxy and others. See the guides for step-by-step setup.