Absolutely. Many, many routers have Ethernet ports and WiFi. Is it possible your router already has WiFi?? Assuming your router doesn't have WiFi, you have a few options. Buy a new router or buy what's called a wireless access point. Although in most cases it's cheaper to buy the router but you never know what you might find on sale.

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.

I've a strong preference to not use ducts (supply or return) so I'd likely just open up the walls where necessary to fish the cable down properly. When I have done that I've used a string (mason's line) or chain w/ a weight on it as my first attempt. Often w/ things like you're encountering you need to bob it up and down, swing it or whatever to get it to go by the obstruction. I've also pushed the weight down w/ a fish stick for better control and then when it get's by the obstruction it will drop on its own. Last option is a fish tape and sometimes again w/ a weight on it. If these are two different ducts then you'll need to carefully exit at the bottom of the upper run, go through obstructions, and then reenter at the top of the lower run. - Make sure to ALWAYS seal up each cable entry/exit very very well. - Plenum approved cable is always a good idea for this type of run but definitely if using a supply duct. - When drilling through between floors be very careful that you don't weaken anything structural. - These can sometimes (or often?) result in noise from cable slap caused by airflow slapping the cable against the duct. Besides the noise it can weaken the cable.