Unfortunately there is no sure way to tell you if you should be using WS Tool or WS View. This is because WS View is only for newer versions of the very same station console. Only one or the other will work. I think it best to start with WS View but don't give up on it too easily, because even for station that really do use WS View there are a lot of details that have to be just right or the WiFi pairing will not work.
If you believe that you have configured the station console on your WiFi network then you should be able to check your router's DHCP leases and see that the MAC address for your station is listed with the IP address that the router assigned it. If WS View is working you should also be able to see the station console under Menu / Device List. Then with the IP address of the console go to a computer and ping it. In Windows CMD prompt you type: ping 192.168.1.131 or whatever the IP address is that you saw. On a Mac open Terminal application under Applications / Utilities and do the same ping command. If you get a ping response then the console is on the network.
As for getting the station on WU and all the other weather services I recommend first creating the station accounts on a computer and then after you have the credentials you enter them using the WS View app (or WS Tool if that is the one that works for your model.)
As for upgrading the station console firmware this is done by the WS View (or WS Tool) app. If there is a firmware update then it will show you that as an option to perform. But the station must be online and connected with the app as this can't be done offline.
There is also the possibility that you have a station console that does not have WiFi. There does exist an identical console without the WiFi capabilities. For example the Ambient WS-1900 is such a thing even though it looks identical to those other consoles that can connect to the Internet. Just something to keep in mind if nothing you do gets you online. If this is the case you can go to Ecowitt and purchase the display console only with WiFi capability and then use the new display to get online and you can still keep the old display as just a secondary display for perhaps a different room just to see the live sensors.
For further troubleshooting if your station is not connected to WiFi yet then I recommend these tips:
https://www.wxforum.net/index.php?topic=38911.msg400102#msg400102