Can I run a wire from the back my ethernet patch panel to my router?
The patch panel usually supplies outlet jacks in various locations in the house - thus all the wires running into the wall! That being the case you feed the FRONT of the panel with ethernet from a switch. The switch is then connected to your router. Thus each jack in the house will then have get a pathway to the router. (You don't connect the router to the back of the panel! It goes to a switch which connects to the front of the panel.) Moving the "hot" connections on the front allows you to make the corresponding jack in whatever rooms you want a "live" internet connection. (Or even phone.)
A single cable from the router cannot server more than one Ethernet port at the same time. You need a dedicated port for each computer/device. Most common solution is to run the router Ethernet cable to a network switch, and then plug all your devices, one at a time, into a dedicated switch port.
This would be a great place to hide the modem/router. If you aren't prepared to move it, then put a switch here to add live ports from that single line. Each wall jack that you will be supplying internet to needs its own port.