Plug your computer into another computer and have them both create an APIPA. Then have them ping each other and see if it drops. Warning, if your NIC is sending back voltage, this might burn the other computers NIC.
Buy a small switch, not a hub, and connect that to the router and computer (you will need another Cat5e cable for this). And see if you still lose connection. The switch should come with a console cable, if not, get one. Connect the console cable to the switch (RJ-45/computer connector) and to your computer (a serial port/ the only part left to connect). If your computer doesn't have a serial port, you can get a USB to Serial cable/converter; you will need the software for that to work. Download a program like hyper terminal or putty (if your computer doesn't have it already). Open up said program make sure you are connecting thru the COM port (if it shows multiple, you will need to trial to see which one it is on) and use Default Settings. Pressing Enter/Return on the next screen should come up with a prompt that will look like: "Cisco>", just without the quotes. If is shows, "Username:" commonly the name is cisco (check the manual if this doesn't work), then it will prompt "Password:". The password is either blank, or "Cisco"; note that you will not see an indicator letting you know that the letters have been accepted.
Then type either "en" or "enable" and it will prompt for a password, same as before. Then type in either "Show Logging Buffer", "sh logging buff", "Show Buffer", or "Show Buffer Logging". And it will display a bunch of lines with info. If you can't figure out what it says, copy the line and paste it into google or on Cisco's website (Cisco.com) and see what it says. Also, if the commands don't work, go to Cisco and see what the command is for your switch model. The model number will look simular to WS-C1224-X. Also look up how to view errors and turning on/off debugging.
That will allow you to see what the switch/router sees, so you will be able to better tell if your NIC is sending back voltage. If the switch is a POE or PWR, then the voltage won't do anything to it. If not, the switch may keep rebooting or the port will eventually burn out.