This escalated quickly!! LOL!!! The obvious solution is to consult with a home theater installer. And I just know I will either end up going for my lungs OR will pay a ton and nothing will work as I specify. I know this. I can buy the Denon and get the 5.1 and the 2.1 set up going. If it comes down to it I could always do a small cheap amp for the kitchen speakers or Heos wireless speakers which would be more expensive. Thgen if I were to do a TV in the living room I'd have to look into powering a 5.1 system there. But thats not for right now.
There's two ways to a approach this. One is to get an AVR that actually does this (I can't find one that does ALL of this). The other, is to use switches and expansion gear, to take one signal and split it up (but with fairly good quality) and then deal with the line level signal in each room from one primary source.
If you use something like a Denon X3700H, it has zones 2 and 3 pre-amp output. For the primary room with the 5.1 surround, this is where the X3700H would live and setup as 5.1. the X3700H has two extra zones, Zone 2 and Zone 3, which are pre-amp outputs (stereo). You will need to double one of them. For the room you intend to use as the guest room and kitchen, double that one; you would use a Y-cable to split the signal (each channel) from 1 to 2 (resulting in 4 new outputs from 2 channels). So zone two can be output to the room you want to eventually do 2.1 in (for this room, you get a basic amp that will power speakers). For the zones 3 and 4, you use those Y split cables. You'll need an amp for each room to receive this signal (you can probably live with 35~50 watt range amplifiers with plenty of room to spare).
That's $1k~1.2k for the AVR (X3700H or equivalent) plus at least 3 amps (let's just use something basic like an Audiosource AMP100 50 watt stereo amp for $125 each). That's under $1800 USD for a source/amp for all 4 rooms. Buy a few y-cables (RCA male 1x to 2x female specifically). You'll obviously need really long rca cables to go from your pre-amp posts to the amp's input posts (unbalanced RCA).
That's your whole budget. No room for speakers, no room for a sub, unless there's more budget. Your source system alone is going to cost at least $1800 plus cables. This leaves $1200 roughly for speakers and subs for 4 rooms. You'll be scraping the barrel for speakers at this budget points times that many rooms.
Very best,