Hey there, glad to have you!
I'm just going to comment on some things, and we'll see for far we'll get before other chime in.
You say you have a Photon, yet try to do troubleshooting steps for a Core. That's not going to help.
You have either not installed DFU-util (properly), or have not added it to your path. Take a look at this guide.
Has your desktop got a wireless network card? It needs to have wifi in order to detect wifi.
That is correct. In order to be able to claim the device, it needs to be online (breathing cyan).
The Photon only supports 2.4GHz. Trying to set it up on 5GHz will not work, and might lead to the problems you're currently facing.
They certainly have nothing to do with your router, so that manual is pretty useless 
The LED colors for the Photon are describes over here.
So, with that said, could you try to get DFU-util installed properly. Then, put your device in DFU mode (blinking yellow) while hooked up to your computer. In the CLI, do particle update, after which you place it in DFU mode once more. Then do particle flash --usb tinker. Once that is done, put the device in listening mode. In the CLI, execute particle identify, and copy the ID. Then do particle serial wifi and follow the steps. Put the device in listening mode and do particle device add [ID]. It should not be online and claimed.
Please don't wait that long, you'd be surprised how quick this community is to react