While the profile is installed, all device traffic goes through the proxy. If the proxy app isn't running on your Mac, the device has no internet. 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.
Yes. Open the same profile link in Safari on the device and install the profile from Settings. It works the same way as on Apple TV — no manual proxy setup needed.
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.