FPGA - Data transfer via Ethernet
I am not familiar with your environment (Nexys 3, etc), but I think the following should be correct in general. When you add the IP core, it creates a block of logic inside the FPGA with the ports being the interfaces internal to the FPGA. Most of the interfaces, you need to connect to internal logics that you define. Parts of the interfaces, you may map to actual external I/O pins. For the external controller LAN8710 on board, it duplicates the function of the IP core with the added necessary physical network interface. Either the LAN8710 or the IP core only provides you with the lower, hardware layers of the networking stacks (the IP core probably does not include the physical layer/interface). The next level would be the TCP/IP(Internet Protocol) layer (assuming you want a standard stack), which is usually implemented in software of a processor.

Therefore, the IP core or LAN8710 after integrated with your logic (not trivial) still leaves you with big gaps (e,g. the TCP/IP and higher layers) with what you want to do. Instead, try the USB-UART idea. Or try an external UART to Ethernet or Wifi converter module.