Hi @doctorb 

Welcome to the Sonos Community!

By a large degree, the easiest (and cheapest) option would be to have each TV-connected Amp located in the room in which it’s delivering sound - and near the TV. In-ceiling speakers aren’t really recommended for Home Theatre, so you may want to look into Sonance In-wall speakers, which are also flush-mounted, to use instead. You’d need two Amps for surround sound, for each room. You could still put the music-only Amps in the media closet and use in-ceiling speakers for those.

I say this because the alternatives are:

• Long HDMI cables - these are expensive and only operate up to 50ft. Also, you don’t want to use them.
• HDMI Matrix switcher, outputting to HDBase-T - this is good if you want to be able to play one source on all 7 TVs, but an expensive option. It would require dedicated ethernet to each TV and a HDBase-T receiver for each TV too. You’d need a Matrix that supports HDMI-ARC being passed back to the Amps next to it (I have no idea if this exists). If you plan on using Smart TV apps for content, this would be overkill.
• HDBase-T to each TV. As with the Matrix switcher, this requires dedicated ethernet to each TV. This time, each TV and each Amp would need a transmitter/receiver. These are for moving AV over ethernet and as you’re only needing audio (that you’ve said), this is also overkill. There’s a thread on this here: 
• HDMI-over-Ethernet (also called HDMI-over-IP). This might be your best (and cheapest) option if you really don’t want the Amps in the rooms with the TVs. Again, each TV and Amp need a HDMI-over-IP unit and an ethernet connection. Instead of using dedicated ethernet, however, this tech can utilise the existing network cabling to route audio (and video). Each unit is also much cheaper than HDBase-T options. It would increase your network traffic, however.

Given this, I recommend Amps with the TVs (mounted behind them, perhaps?), but if you really don’t want that then HDMI-over-IP is probably your best bet. You could even have a second router running that network, keeping the cables separate, and thus keep this extra bandwidth off your actual network. It shouldn’t be too much, however, as only audio is being transmitted. I’m not sure how you’d connect these up - they are intended to carry audio and video, but not necessarily ARC which goes in the opposite direction and using a different wire on the cable. I recommend plenty of research.

There may be other options I haven’t thought of.

Really, this looks like a job for a professional AV guy.  We can give advice, but the layout of the different rooms, as well as discussions on how they will be used, will narrow down how best to handle that room.  You may want to use ceiling speakers for some, or use a Beam or Arc for others.  You may want to use a speaker selector and combine some of the rooms into one zone, running off one Amp.  You also may want to prioritize which rooms you do first, or setup guest rooms on a different network etc.