Yes to both, actually.
Yeah, I knew I wouldn't be able to get away with just a single receiver. My question was what equipment I would need to get. Like, is there some 1RU box that I can buy that would allow for that sort of multiplexing? Do I need to buy a box for each different set of controls? Those sorts of things.
First consider this. If you use a central AVR, you are limited in how much control you'll have from each room. It is why many people opt for a separate AVR in each room. You can do it on the cheap because you don't need much power or many channels.
Speakers in multiple rooms wired to the same source can be useful for something like background music in a open house. For family use and flexibility, not so much.
The only way I know to get more than one source at the same time from an AVR is by using Zone-2 and Zone-3. I think the newest high-end AVRs allow you to use HDMI inputs for the Zones. So you could run one pair of speakers to Left/Right, one pair to L/R Surround and configure them for Zone-2, and one pair to L/R Rear Surround and configure them for Zone-3. You could control source and volume for each room from the AVR.
But if you're in the Zone-2 room and want to turn the volume up, or change the source, you have to go in the other room to the AVR. Walk back and forth until the volume is the level you want. Or you could put a volume control in each room, but you'll have to find the speaker wires in the room and pull them to that location... (the same thing you'd have to do if you add an AVR in each room).
Think about what you want, and people here can give some equipment suggestions.