> development of a loopback code for this processor (either using sockets, ARP request, or anything else)
All of those protocol levels assume you're connected to a network, and connected to something else you can send addressed packets to
To do what you want you have to send and receive raw ethernet packets.
You didn't say what software or operating system or network stack you're using. It may have a "send raw packet on ethernet" and "receive all packets from ethernet" interface function. If it doesn't, you'll have to hire a programmer to write one.
You're not testing it properly with loopback. The hardware may have a fault where it can talk to itself, but not talk to anything else. Ethernet requires tight tolerances on the crystal controlling the Ethernet timing. It is either "100 ppm" or "+- 50 ppm" (refer to your data sheets or the relevant Ethernet standards documents). If you're not separately measuring the generated frequency against an external reference, then it might be wrong. If it is a long way out of spec (10000 ppm or more) then you can't exchange any packets with external devices, but it WILL pass loopback tests. If it is over 100 ppm but less than "a bit more than that", you can send and receive short packets, but not long ones. If you're directly connected to the test device you can be a lot further out of spec than if you're sending through a switch or hub, so you should try and test through a hub or switch with maximum-length Ethernet frames.
Tom