I'm not quite sure how to do this.
Will the NAS keep the ip it had when connected to the router when I connect it directly to my laptop?
Looks like my laptop got a different ip than when it was connected to the router. So how can I find the new NAS-ip?
First off even though your DNS-323 is brand new have you tried resetting it to factory settings like was suggested in the link you found (http://forums.dlink.com/index.php?topic=11325.0)? The reset button is a on the back, small hole (use paperclip, maybe a pen) between the power cable and the Ethernet connector. Just because it is new, doesn't mean that there is no chance of this working. Especially if you ran the "Easy Search Utility" and something went wrong. BTW for the first try after this I would tried it without the drives, just in case.
Speaking of which did you run the "Easy Search Utility" or just jump to using the web interface?
Whether it works or not might tell a lot.
Personally I think it is highly unlikely this is a web browser problem, and as you have found out there is no way to update the firmware without being able to log in, and for Pete's Sake, it doesn't matter what firmware version is on the DNS-323 they should all allow you to log in or it would be totally useless.
And I can't see how a router would be blocking just this one action.
To clarify, I use both Firefox 3.6 and IE 8, with no problems And I can tell you that you disabled Javascript (which would have taken work on your part) on the login screen the Username and password prompts would be on a gray background instead of a while box background, and when you selected the Configuration button it would do nothing (as in it wouldn't go to an empty page).
On the connecting the DNS-323 directly to your computer. It certainly will behave differently. Your router was most likely assigning it an IP address, and that will be gone without it. From the manual DNS-323 first tries to get an IP address automatically and if it can't then it will use 192.168.0.32.
Your PC is not setup to assign an IP address to so that is what it will default to, but that brings up more of another problem. Your PC is also getting its IP address from that router. If you disconnect you PC from the network/router the PC will assign itself an IP address and if it does it the IP address it assigns will not be on the same sub net so the two will not be able talk to each other. You have to know how to manually assign your PC an IP address and network mask.
If you know how to do that give it an IP address of 192.168.0.2 and a mask of 255.255.255.0 and it should connect (make sure you use http://192.168.0.32, and give it a few seconds after connecting the cable before trying). If you don't maybe someone else can give you the instructions. I know them, but I think this line of troubleshooting is more likely going to cause you more problems then it solves.
I would think you should try the reset, and if that doesn't fix it the "Easy Search Utility", and if that doesn't work return it and try another, or if they provide it call D-Link.