Running at 3Mb/sec on a 10/half duplex connection, and not being able to connect at 100Mbit can mean 2 things: 1 either the router, or your network card doesn't support 100 Mbit, or, and this is the most likely cause, your cable/connector bad. I would therefor suggest using a different, new cable.
To the person with the modem - router - pc setup:
First check if the modem is really setup as a modem, and not a router. Meaning: check the external ip address of the router, this should be a public addres, and not 192.x.x.x or 10.x.x.x
If the outside connection of the router is something like the above addresses, make sure you don't use the same range for dhcp to your pc. as an example: if your router has an external address of 192.168.0.2 the inside dhcp pool should not be 192.168.0 but f.i. 10.0.0.x
Also a good test would be: disconnect the router from the internet modem, and turn on the pc. Do u now have a connection to your router?
Now reconnect the modem, can u still reach the router, and if so, does the router get an ip from the modem?