Seeking flexibility to mix and match video and audio sources

Hi all--I hope this friendly forum can offer some advice on choosing a receiver. I'm novice when it comes to modern networked AV receivers.

In addition to standard living-room use cases (TV/movies/music), I'd love to have a receiver for more sophisticated/flexible scenarios where audio and video signals come from different inputs and it is simple to switch the video output between my television (via HDMI) and projector (via composite/component/VGA/"old stuff"). Here are some example situations to help illustrate what I'm looking for:

1. HDMI video coming from a computer routed to my TV via HDMI with audio coming to the receiver via RCA cables (shortwave radio, old tape player, a small analog mixer, etc.)
2. video from an AppleTV over HDMI going out to a projector via composite or component paired with audio from the built-in FM tuner.
3. video from an old clunky VCR paired with audio from an AppleTV

Do consumer-level receivers that do such things exist? If so, what would you call such a feature and/or what models are available? Any tips from you experts would be much appreciated.

::crickets::

Ok, I'll take that as a no. :-)

A simpler question: can your average receiver take a video signal that comes in digital (over hdmi) and send it out as analog (over composite or component). What feature do I look for to ensure it can do that?

Ah, thanks @BIslander. I never thought about the copy protection aspect of "downgrading" the signal like that. Makes sense.

Yes, I'm starting to download PDFs of manuals from various manufacturers to try and get the full scoop.

If anyone has knowledge of specific models that allow the mix-and-match of audio and video sources, that would be a great starting point for me.