The use of a microphone through the system complicates things and you really should not be looking at residential product, but commercial product.
That is, you need a mixer of some sort, and that mixer could be a standard mixer (microphone in, source in, single output) or it could be a matrix audio mixer which allows for different zones to play different things at different levels. I actually don't work with standard mixers much, but instead use digital matrix mixers which provides similar functionality in less space, but needs to be programmed.
If it were me though...
Polycom Teleconference Vortex EF2201 and EF2280 | eBay
That gives you up to 12 channels of audio input (6 stereo pairs). More accurately, up to five different stereo sources could be put into it, and two microphones. It includes phantom power to run the mics.
Then, the 12 outputs allow for up to 12 individual mono rooms to be run. Yeah, complete overkill, but for $80US (right now) with power supplies and a backup, it's a great price.
So, all your sources run into that unit, then the output would go to something like this:
Crown cts 8200 Professional 8 Channel Amplifier No Reserve Don'T Miss It | eBay
This will get you amplification for 8 ohm, 4 ohm, or 70v/100v speakers in all your rooms from a single amplifier which is designed for long term use in less than ideal conditions. Each room will be individually amplified and can have its own source, or a combination of sources as desired.
Volume control for individual rooms, at this point can happen in one of several areas. First, you can adjust the gain on the back of the amplifier for each room/speaker as desired. Or, you could use the digital matrix mixer and adjust sound to each room as appropriate.
If that's all to complex, you could then add volume controls in-line with the output of the amplifier to adjust volume for the rooms.
If it were me, I would pick up a used Crestron control system and a touchpanel and put some mic level and program level, and source control right onto a visual touchpanel interface which would make it easy to change and could be password protected.
Crestron MC2E Compact Control System with Ethernet | eBay
Crestron CNPWS 75 75 Watt Cresnet Power Supply | eBay
Crestron TPS 6L | eBay
From the amplifier, speaker wire out to your speakers. There are dozens of different types of speakers out there that run all over the place on pricing. Generally, you will get what you pay for and since we aren't there, I'm not really going to specify a on-wall type speaker which is appropriate for your setup needs and budget. But, the amplifier will definitely be able to handle up to two 8 ohm speakers per channel for up to 16 speakers driven, or many more depending on how you wire things. This system will be robust ensuring long term reliability, and depending on some of the options you throw in, it will be as easy to use as you want it to be.
