Every user in our company needs to use GuiXT. Does this mean GuiXT has to be installed on every single local PC, or it is possible to install it on SAP application server, which would be my preferred solution?

GuiXT has to be installed on each and every PC, like sapgui. In fact the normal sapgui installation already contains guixt.exe and guixt.dll.

Only in the case of connecting through Browser/ITS (sapgui for html), however, is GuiXT installed centrally on the ITS server.