Mine does much the same - uses the default gateway as the DNS server - I suppose the router is clever enough to cope with that. I don't use DHCP because there's a VPN into this machine and that needs a fixed IP address, but it looks like this:
Ethernet adapter Local Area Connection:
Connection-specific DNS Suffix . :
Description . . . . . . . . . . . : Realtek RTL8139/810x Family Fast Ethernet NIC
Physical Address. . . . . . . . . : 00-0B-6A-92-30-A6
Dhcp Enabled. . . . . . . . . . . : No
IP Address. . . . . . . . . . . . : 192.168.8.110
Subnet Mask . . . . . . . . . . . : 255.255.255.0
Default Gateway . . . . . . . . . : 192.168.8.1
DNS Servers . . . . . . . . . . . : 192.168.8.1
What does seem a tad odd to me is that this machine is 10.0.0.1 - somehow, I'd have expected .1 to be reserved. My daughter has a similar set of addresses, and her first machine is 10.0.0.4 - other machines work up from there.
Anyway, let's see how Juud makes out from this step. I'd have expected Auto DHCP to work just fine, but if it won't, then fixed IPs do have a certain stability to recommend them.