Hi volodia82,
Welcome to Apple Support Communities. I understand your iPhone is unable to create a screen recording, along with the inability to connect to your Apple TV. I'll be happy to help.
If you haven't tried yet, I recommend restarting your iPhone: Restart your iPhone This can help resolve unexpected behavior. In addition, you'll want to make sure that your iPhone has the latest version of the iOS software. This helps provide software improvements on your device. Here's a helpful article with the proper steps: Update your iPhone, iPad, or iPod touch
For Apple TV not connecting to your iPhone, please be sure to follow the steps outlined here: Set up the Apple TV Remote on your iPhone, iPad, or iPod touch The section that says, "Get help" provides additional steps that can help resolve the ability to connect the iPhone to your Apple TV.
Kind regards.