True Copy jobs should not require a computer or care who is logged in. You might be accidentally selecting a scan to email/scan to folder function instead which is dependant on computer connections.
Its a good question if the printer software needs to be installed for each user or not. I would assume it will install for all users but perhaps not. Honestly do not come across that scenario very often. There should be no harm installing the printer as a different user though. If anything the install will go very quickly since all the files are already on the machine.
One last comment, if you have access to a Ethernet cable to connect to your computer then you should use it. I would always connect an Ethernet cable between the printer and router. What you do between the router and the connected devices is up to you. Ethernet cable connections are much less complicated and are not prone to falling asleep or dropping IP Addresses. Wireless only really shines when it is not possible to use a Ethernet cable (tablets, smart phones, no physical routes for a cable) or the PC is highly moble and rarely in the same spot when used.