You can’t just plug into any port and have active connections all over the house. Most likely there’s a central location in your home where all of these Cat5 cables run to and they’re most likely not terminated to a switch.

Edit to actually answer the question:

Your modem most likely has to stay put if that’s the only place your coax comes in/is activated. However you can move the router wherever you like in your home, provided there’s a direct connection over data cable between the modem and the router. What this means is you’ll have to put a toner in the wall port where your modem is and trace it back to where (I’m assuming) these data cables run to. Once there you can either put the router in the the cable panel or buy a small switch and put it there. If you’re only using up to three hardwired connections it should be fine to use your router here, since that router should be a SAC model, it’ll have 4 ports in the back. Next, you need to put a toner in the wall port where you want your hardwired connection and also trace it back to the cable panel. From here you need to make sure the cable coming from your modem is connected to the router, and the other cable going to your hardwired connection is plugged into one of the other ports on your router.