Heh. In my book, no question is unnecessary, if you don’t know the answers. 

So, your original connection (the TV part) is correct. The Beam goes to the ARC port on the Frame’s connection box, the Apple TV goes to any other HDMI port on that box.

If you’re using the Amp to drive a set of surround speakers, then yes, it would connect wirelessly to the Beam. Sonos engineers apparently call it via SonosNet, but I object to that appellation. In any case, it’s a ‘hidden’ 5Ghz connection, and it’s called ‘bonding’, which places both the Beam and the Amp in the same Sonos ‘room’, so they’ll play at the same time when using the TV input.

That being said, you could also choose to set up the Amp separately, in which case it connects to your router, either via Ethernet, or your WiFi signal. You could then group the two ‘rooms’, and play streamed music in sync, but the Beam’s TV input would be delayed on the Amp’s ‘room’.

It depends on what you want. My recommendation is the former, but there’s nothing wrong with the latter, either. 

This ‘may’ be helpful, if you’re interested in Sonos’ jargon, which most of us use:

Sonos Terminology

Not sure why it isn’t an FAQ that is slightly easier to look up, but there is a lot of good information there. 

Thank you! This is all very helpful.

I’d like to better understand the TV audio sound delay between the Beam and the Amp — if they’re setup in separate rooms and grouped together, rather than being bonded in the same room.

Could that delay be minimized if the output from the Frame TV was connected to each one independently, i.e., the Beam (via arc HDMI) and the Amp (via plain old HDMI)?

Or perhaps if the optical output from the Frame TV was split into two paths, i.e., the Beam (via HDMI converter) and the Amp (via HDMI converter)?

No. 

The ‘delay’ is the way that Sonos works in software, while there is some machinations you could do to make the  rooms closer, you’re essentially delaying the ‘first’ room, so that your lipsync would be off further on the TV. You’re much better off ‘bonding’ the two devices, IMHO, in a single ‘room’, and using the speakers connected to the Amp as surround speakers….depending on location, of course. In that case, when playing music on the Beam via a stream, you can go in to the options, and set up the ‘surround’ speakers as ‘full’, rather than ‘ambient’, and they’d operate as a stereo pair, rather than merely ambient support for the Beam. 

All that being said, if they’re in two rooms, you wouldn’t hear both sets of speakers at the same time, so setting them up as two ‘rooms’ in the Sonos software makes sense. You’d then have full separate control over volume in each ‘room’ (zone), and could choose to group them, or play different sources. They’d connect to your WiFi in that case. 

I used to have a pair of speakers in my kitchen, ‘grouped’ with my TV speakers. Since I could only hear both sets of speakers when standing in the doorway of my kitchen, it wasn’t an issue, and I could still cook while listening to the football games. Wouldn’t work for me now, as I have an open floor plan, but I just turn up the TV a slight bit. 
 

Oh, forgot…sorry. The HDMI ARC shouldn’t (normally speaking) be split, without relatively expensive equipment to boost the power/signal. Even if you were to convert the audio signal to optical, I’d still recommend a powered device (less expensive, but still not zero).

If the Amp is in another room, why do you care if there is a 75ms delay in the audio?

Ironically, my situation is very similar to yours — an open floor plan with connected rooms. And I’d rather not have a soundbar blaring in the distance on game day if I can help it!

If there was some way for me to ignore the echo that comes with a small delay…believe me, I would. But even the smallest echo drives me bonkers

As I mentioned, this is all new territory for me. Most receivers I’ve seen take a feed and play it across all zones in harmony. So this is certainly a unique situation — but I’m sure there’s a good reason for Sonos doing it this way. 

Just to confirm: even if I were to split the optical feed from the TV into 1) the Beam (via HDMI converter) and 2) the Amp (via HDMI converter), there would still be a 75ms delay between them?

No. If you fed the feed from the TV to both devices (not sure this is feasible, I’m speaking theoretically here), there should be no delay between the rooms, as they’re not grouped , and technically unaware of each other, so not subject to the Sonos software ‘buffering’ (there’s more than that going on, I suspect) between a home theater ‘room’ and a grouped ‘room’.