Fortunately for you, I have the very same adapter and I can share my results with you.
For me, the device and ethernet connection has performed perfectly. For starters, it would be interesting for us to know what you are connecting to. Is it ADSL? Cable? I know it's not common to reveal too much of yourself because of hackers and stuff, but it's rather pertainant in this case because it makes a difference.
I am connecting to AT&T broadband cable. I have a Linux box acting as my router (NAT server) which also serves up DHCP. My DHCP server is tweaked up nicely and obviously works for Windows and Linux machines alike. So my WinCE device is just another source of IP traffic. My Linux router doesn't care.
However, I have seen that various cable internet implementations rely heavily on host/DNS information. Additionally, and especially in the case of AT&T broadband, until your ethernet device is "registered" with their routers and stuff, you will not be able to get real access. You will not be able to ping or anything. So you will not be able to see anything like communications to your host.
The work-around, if you are an AT&T cable subscriber, will be that you will have to install the USB ethernet device on a Windows machine and get it registered properly. Once registered, you should be able to move the ethernet device back to the IA-1 and actually get online using it.
If you're using another service, I will tell you what I know and that might clue you in more. I have already written to MSN's support on the matter of their Broadband implementation and the obvious pitfalls in their too-simplistic approach. Since there isn't any host name information stored or managed, you're just an anonymous "IP" host...cable doesn't like it, though DSL is often okay with it like that.