You seem to be having an unfortunate set of problems with your board. I probably would have been pulling my hair out if my EVB gave me this many headaches.
OK, so, I'm assuming you JUST want to get Linux running (flashed) into the board for now...
I have had no issues with various interconnect topologies. I usually have the PC on a Netgear Gigabit VPN router, connected to a Netgear Gigabit switch (for port expansion), connected to the EVB. I've got 3 PCs (2 Windows, 1 Fedora8), a NAS, a networked printer, a wireless access point, some other development boards ALL connected. I figure if anything would cause the EVB problems, my setup would be it!
The main difference is that I use a Windows TFTP program to transfer the image to the EVB (SolarWinds.net TFTP Server - free version). That works fine with U-Boot. I've loaded many builds that way (both later programmed to flash, and executing from SDRAM for testing).
Remember U-Boot isn't running DHCP (uses FIXED addresses set in environment variables). Make sure the network U-Boot environment variables are set correctly (and make sure to SAVE them before you power cycle). You can probably setup your router to make a STATIC route between your PC and the EVB. Remember, the Linux PC is probably going to request a DHCP address from the router (unless you set it to a static address too). So, addresses MIGHT change (and break your static route). Usually, unless you move cables around, router DHCP tables seem to stay fairly constant though.
I have successfully TFTP connected using Fedora8 (should be very close to your Red Hat kernel), but usually transfer EVB builds from the Linux machine to the XP machine for downloading (since I have CodeWarrior on that machine as well).
If you were getting timeout errors immediately, I'd say your network variables were probably not set correctly. Since you're getting checksum errors, it sounds like you're connecting but having other issues? Anyway, check the "gatewayip", "netmask", "ipaddr", and "serverip" addresses to make sure they match your network setup.
Good luck! Hopefully the board will work for you soon. Once it's running, it seems to work pretty well. I routinely xfer files via network to the EVB's hard drive and/or use a USB flash drive.