Yeah you can have a LAN cable going out and one coming in i think NVIDIA use it on their motherboards or something so theres a cable for outgoing and one for incoming its a good idea and its supposed to reduce your ping
Yes, you can have two cards in your PC, yes they can both be connected to the same internet connection but this will not increase your performance in any way.
If you are only using one internet connection through a router then if you connected your PC with two Ethernet cables to that router, the internet is simply being divided by 2. However, the PC wouldn't use both simultaneously. It would just choose one of them...if one stopped working, it would revert to the other.