No, lacking DNS is not in itself strange. The strange thing is how this lack arises.
See, every time I send a DNS query packet from one of the several computers behind the router, what comes back is a response packet that purports to tell be the IP address of www.kimo.com.tw. Apparently the router is not trying to falsify the address of the host I want to find. I ask about, for example, www.google.com, and back comes a packet saying (translated from RFC-1035 speak): "Thank you for your inquiry about the IP address of www.kimo.com.tw. It is my pleasure to inform you that the IP address of www.kimo.com.tw is 22.214.171.124".
It's always www.kimo.com.tw. It's always 126.96.36.199. It only happens for UDP queries; DNS over TCP is unaffected. It's not the nameserver at my ISP that misbehaves; I get the same pattern when I ask a root server about "com.". I don't know whether the router rewrites my outgoing requests to be about www.kimo.com.tw, or responds with a stock reply on its own, or rewrites incoming answers. I don't know whether it affects UDP packets not to/from port 53.
This has happened two or three times over a period of several years. It seems to tend to follow downtime on the ADSL connection. But whenever it happens, rebooting my local router clears the problem. It's very strange.
Is the router getting infected with some malware? I have a hard time figuring out what said malware could be attempting to achieve. Because the name in the reply does not match that in the request, the resolver on my local computer will just fail instead of return a wrong answer to the application.
After intensive web searching the best I have been able to find is this page in Russian, which judging by Google's translation seems to describe exactly this syndrome. Apparently it is claimed that no malware is involved, but I cannot make sense of the machine-translated explanation of what actually happens.