Since this is a Sonos forum, I’ll give you the Sonos answer, for answers to 2 and 3, you’ll likely need to look at another forum for non specific manufacturer answers, such as AVS forum or the like. Alternatively, you may want to engage a professional installer, who might give you advice based on what they know, sell and install.
If you read the specs on the Sonos Amp, it can drive up to two pairs of 8 Ohm speakers, or 6 pairs of the Sonance Architectural Series speakers, which are specifically designed for it. Each Amp you would use is its own ‘stream/zone/room’, so you get a single stream and volume out of it. In general, Sonos devices connect wirelessly to your WiFi environment, but the non-portable ones also have Ethernet ports which can be connected (at 10/100 speeds, not gigabit speeds) to your network.
When streaming music to the Sonos system all ‘zones/rooms’ will be synchronized. There will be a slight delay between your home theater setup and any grouped rooms when playing the TV input on your Arc Ultra/Sub4/Era 300s. It won’t make a difference if they’re wired or wirelessly connected.