A single processor will actually do. Having multiple sites opened (tabbed or new windows) only takes up some memory, but not that much to slow your computer down. I'd recommend at least 2GB memory, but the more the better - though the faster you make your netbook/notebook, the faster you'll drain your battery.
I'm not sure how much battery life is important for you, but if eye strain is a problem, making your screen brighter when on battery power will drain more battery. Plus, the larger the screen size, the more battery drainage.
Reliability is a tough one, you'd just have to read the user reviews at different places and see what everyone says about it. For compactness, I wouldn't upgrade to a 9 or 12 cell battery, just stick with the 6 cell battery. Your battery life won't be as long, but your netbook/notebook won't be as heavy. Good operating speed would be a single or dual core no faster than 1.8 GHz. I'd recommend a 64-bit operating system, but then you'd need at least 3GB memory, which you don't need. A 32-bit OS with 2GB memory should be fine.
Anything up to 15.6 inches, you'll get resolutions up to 1200 x 800 (14.1"). I'd recommend using your desktop monitor and change your resolutions to get a better picture of what those resolutions look like with your new netbook/notebook. If I need more vertical viewing space, I just press F11 to maximize the browser reading space.
The bigger the monitor, the more battery drainage and heavier the netbook/notebook, but the more comfortable the viewing and viewing angle.
The smaller the monitor, the less battery drainage and lighter the netbook/notebook, but the less comfortable the viewing and the viewing angle. I notice I have to slouch more or sit up straighter on the 14.1" notebook than my 17" notebook.