Question about Apple HomePod 1st Gen.

Watch videos via AppleTV4. Audio via AirPlay to the HomePods (I have 2), video via HDMI cable to TV. Works perfectly. Apps in AppleTV must, however, be enabled by developers themselves.

With Apple TV, the connection via AirPlay 2 should not be a problem. As far as I know, TVs do not have AirPlay support.

Unfortunately no, it does not have an incoming connection

The connection to the Apple TV works with the Homepod. The Homepod must simply be selected as the output device via Airplay 2.
Caution: The picture is delayed in order to be in sync with the sound. This means that "gaming" on the Apple TV is no longer really possible while connected to the Homepod.

Directly from the TV does not work
From the Apple TV it works without problems

For pairing with Apple TV, I would say this should work. However, I don't have an Apple TV. Only the Homepod.
Greetings