So, I will ask before continuing that if there is not a different solution than that outlined below, please just tell me how to factory reset.
To summarise the first issue, I am in a new location looking to set up the wifi. Now, I cannot set it up via the router as the SSID cannot be found (or does not match the computer) and so, the details on the printer and my computer are perpetually different. The printer is, for some reason, connected to an open network that I have never heard of and certainly never connected it or my computer to. This cannot be changed. I have uninstalled the driver and tried setting it up as a new printer. I have reset the wireless (wifi button + cancel). I have tried to set it up completely wirelessly and via USB. I have tried to input the correct information completely manually. The information is just not accepted. I have been suggested to just 'connect it to the correct router' but I hope you can see now how that is just not the case at the moment.
Secondly. I cannot access the settings under the printer 'homepage' (I am aware of how to access the homepage itself, I refer to http://192.168.223.1 or http://[hostname]) as it is locked by the administrator. I set these details myself and unfortunately lost them (I now have the password via the network configuration page, but I need the username) and feel as though I need to access it to help with the first problem. I have contacted the HP support via social media who told me that the login is the router details - Which it is not, and the printer has never been able to connect to this router (see issue above). The username is not admin, nor blank, nor any variation of. The printer homepage tells me the network details (i.e the unknown open source that it is connected to) but I cannot access anything else due to not having the administrator details. I have tried all the solutions I listed above to also try and solve this, to no avail.
Again, unless there is anything else that I haven't tried, please inform me. If not, I think the best solution would be to reset it entirely.