Yes, you need to add the right ip in the db/hosts file, so a hostname corresponds to the ip.
As you can ping your own ip-adress on the Amiga, it sounds like AmiTCP in itself works fine, but there is some issue with the device or something. Have you entered the device name as "FastEthernet.device" with a big F and E in the interfaces file? The case of the letters matter for devices.
Btw, if it is a 100Mbit card, I think Picasso96 needs to be running. Those cards need buffer data in vram, as that is the only place pci-cards (the network card) can dma to/from as the motherboard space is a no-no, and correct me if I am wrong, but I think I read somewhere that they need Picasso96 running to allocate that memory.
/Patrik