Generally speaking, it will be Corp dominate in the form of its companies. From what i understand, Extraterritoriality only extends to businesses with directly under a division corp unless the Big Name label is on it. For example, unless you are in Atzlan (and even some of them are obscure), Aztechnology operates through distant subsidiaries without putting their name on it obviously. As such, they DO NOT benefit from extraterritoriality. Horizon, however, puts their name on EVERYTHING, so they have a freakish amount of corporate land to operate on.
The styles wage-slaves favor are the styles typically sponsored by the corp they live in, but it does depend. For example, Shiawase folks on Shiawase land will probably prefer threadss from Vashon Island (one of their clothing lines IIRC). If someone maintains dual citizenship (Ares and UCAS, for example, as much of Detroit is), they will potentially wear threads from other lines, where smaller companies and corps can compete. But successful (and popular) wage-slaves will likely wear the Victory line of clothes to represent their loyalty and dedication to Ares.
As for how it is different, how is it different going from the Northern US to the Southern Canada? Not drastically different, but subtle changes. Different brands of beer are more popular. Uniforms of the guards trying to arrest you are different, and they might use some different equipment (Horizon guys using gel/SnS/Tasers, while Azzies are carrying everything but the kitchen sink).
And regarding DocWagon, nothing I have seen is set, except regarding Evo, since CrashCart is a subsidiary of Evo. But there were reports of CrashCart and DocWagon working together to cover overlapping areas. DocWagon generally has to request permission from the parent company if extraterritoriality is involved. Usually it is given, then "clients" are turned over to Lone Star/or the invaded corporation if trespassing was involved and the corporation (for whatever reason) allowed the extraction of the DocWagon clients.