Most routers come with a perfectly fine ethernet cable. If you need other ethernet cables to run to individual computers, don't get suckered into buying an expensive one at your local store. A couple of bucks is adequate. Here is one I just found on Amazon that is 25-feet long, has gold-plated contacts, gets great reviews, costs 98 cents, and ships for free! Here is a link that might be useful: ethernet cable

While it works best to have your modem, router, and patch panel all centrally located, if it helps solve your current problem, the modem could be "virtually located" in the mechanical room. In other words, if the modem is currently being fed it's signal in a particular room, and that room has a network jack, you could then send the internet connection down to the mechanical room and to the router. While you are not splitting the signal, in the traditional sense, and it does not weaken, you still have some limitations. The output from the modem goes into the router. The router then acts like a "traffic cop" directing information from the modem, to-and-from the individual ports on the router. The modem has an IP (internet protocol) address assigned to it from your ISP (internet service provider). The router then assigns IP addresses to each device connected to it, and carefully routes data from each port's address through the main IP address and back again - all using just a single connection. The limitation is in how much data each connected device is downloading, or uploading. I'm in a rural area with DSL that has about 1.5 Mbs download speed. To stream something like Netflix in standard definition, that connection speed is about the minimum you can get away with. Can two people watch two separate streams? No. Can four people browse the internet? Sure. We have three computers normally on-line, home security cameras, internet radio streaming and some misc equipment - and no problems. Download and upload speeds for residential service is almost always asynchronous, meaning you can usually only upload at a fraction of your download speed. This usually works well, since that's the way we normally use the internet. However, if you have something like home security cameras that can be viewed over the internet, you going to be uploading a lot of data and possibly slowing traffic on the download side of your internet connection. So, again the limitation is in the amount of data being received and sent. As a matter of fact, most 4 port routers usually allow a second router to daisy chain to it and create additional ports.

if the hole the coax comes down is big enough, the easiest way to get the cable inteh box with teh existing coax requires a 2 pull method. remove the faceplate and attach the ehternet cable to the coax with tape. go inthe attic, pull the coax up until the ehternet is inthe attic. then untape them and pull several more feet of ethernet cable into the attic. tape the coax back to the ehternet cable and pull the cable back down so that the coax is back inthe box and untape it. you still have plenty of ehternet cable up inthe attic to continue pulling to your destination. if you have to drill a new hole inthe top plate, then remove the existing box and put an old work LOW VOLTAGE box in it's place. LV boxes have no back, so you can easily get the cable in it.

There may be some confusion since this there are a couple posts on this same installation. You can not connect a switch directly to the cable modem unless it is a "gateway" - a router is required. Even if it is a gateway they can be very problematic. The router connects the WAN (from the modem) to the LAN. The switch interconnects the devices on the LAN. Most current routers nowadays are optimized for streaming A/V as one of their features. They monitor their own built-in (switch) ports and prioritize data to help ensure smoother media flow. A switch on a router port, feeding multiple ports, will generally not receive this same prioritization. That is why I recommend connecting streaming media devices directly to the router. Streaming media on a home network from NAS makes no difference. The chain should be: cable modem, router (with it's ports feeding streaming media connections), and a switch connected to one of the router's ports, feeding the other network connections, for best overall performance.