Component video cable can be terminated on the ends in the field with just a few tools and connectors.
The cable for running component in the home can be something like Mini-HR cable (mini high-resolution) and it pops up on eBay from time to time.
For example, this is a brand new roll of 500' of MHR cable...
It has 5 mini coaxial cables which work perfectly for component HD video distribution, with the potential to carry digital audio to/from a location or a analog left/right signal since there are 5 conductors inside.
Now, you count your rooms and your sources... Call it 4 cable tuners, plus a few other playback sources which deliver component video for distribution (Media players, older Blu-ray or similar)... Maybe up to 8 total distribution sources for video. You also have a few more than 8 rooms as destinations, looks like you have 4+ TVs in the basement, and another half dozen or more in the home. So, I would be looking for a 8x16 or larger matrix switcher.
This model, is NOT showing the number of inputs/outputs on it, so perhaps message them, but this is the concept:
Here is one which is a full 16x16...
That will do up to 16 different component video sources and distribute it out to 16 different rooms in ANY COMBINATION that you want! So, if you have 8 cable boxes, they could be viewed anywhere in your home that is wired, in any combination that you want to use.
That's under $500 to get this level of video distribution. You will still need terminations for the MHR cable, which costs a couple hundred bucks to get the parts for, but not that bad.
You will still need the audio side of the equipment which means amplifiers for all the speakers (one pair of channels per speaker pair). Perhaps 2 or 3 12-channel amplifiers. About $400 each via auction...
I would look for deals, but wouldn't overlook quality. Niles, Sonance, and Speakercraft are all excellent manufacturers of distributed audio amplifiers with a long history of support and quality. My personal favorite is the Rotel RMB-1048 which has almost zero noise on the line, but is a bit pricier and more difficult to come by. Call it $1,200 in amps.
There are other options on amps, and a fair number of audio preamps for speaker distribution which are available. I use Crestron for all of it since I can program that stuff. I can get a Ethernet controlled processor, enough to do 16 zones, along with the preamp distribution for about $1,000.
There's still a lot of other costs and considerations, and you need hundreds of feet of speaker cable (14/4 & 14/2) at least. Maybe $40 in speaker wire per run (give or take), so perhaps $800 in speaker wire.
$300 in video switching
$500 in connections for video switching
$1,200 in amplification
$400 in video cable (high side)
$800 in speaker cable (high side)
$1,000 in preamps/audio distribution/volume control with Ethernet enabled control system
That's $4,200 and I would plan on running a lot of cat-6 cabling throughout the house, which will run you several hundred dollars for 3 or so spools of 1,000'.
That'll get you wired up, and take care of your head end.
You will still need to purchase some speakers, you will need to take care of wall plates, power, and the like. You will need TVs and such all in place, wall mounts, etc. So, there is a LOT of additional cost beyond this, but this would be a very top shelf component video distribution system that would have run, rather easily, in the range of $25,000+ if purchased new.
That 16x16 switcher is still sold new in a very slightly upgraded version, and carries a MSRP of over $17,000! Think about what that means in terms of what this represents for quality for your money.