Please, note that this forum is for support related to the Cool SL Viewer only. Questions related with Linden Lab's official viewer should be directed to Linden Lab, via they support ticket system...
ATI graphic cards (and more exactly, their driver) are known to have issues with OpenGL games (and not only SL).
But you should nevertheless be able to use ATI cards to connect to SL, provided that you turn off the "AI" feature in the ATI driver (somewhere, most probably in its advanced settings). Also, some ATI cards/drivers are known to have broken VBOs: you can turn VBO off in the Preferences floater, Graphics tab, "Hardware settings" sub-floater of the viewer (uncheck "Enable OpenGL Vertex buffer Objects").
NVIDIA graphic cards are the best choice for OpenGL games/applications...
I didn't hear of any incompatibilty, but I can't say for sure (Linux, here...).
Yes, but it won't be of any help if you can't connect with the official viewer in the first place. In fact, it's even a bad idea if you want to open a ticket, since Linden Lab would then close your ticket saying "we don't support third parties viewer", not even caring that this is not the actual problem... So, my advice is to first connect to SL with the official viewer, then only once you have been successful doing so, you can install the Cool SL Viewer diff files package over it (see the
Windows installation FAQ, please). Note also that you can install several different viewers on the same computer (simply install them in different directories).
This question should be directed to Linden Lab, but I don't see any reason why you won't be able to connect. To get further hints about what is going wrong, please have a look at the log file the viewer provides (see the
Windows FAQ for where the logs are located), and do attach it to any ticket you would open.