My geiger doesn't seem connected

You should be getting some debug text from the arduino when it starts up an is running. With everything connected up, power the arduino from your computer and run the arduino software. Then open a serial monitor at 9600 baud rate and let it run for a couple of minutes. You should eb getting some messages back from the ardiuno that might help in what is happening. Try and get some debug messages and copy/paste them back here.

Ahh, nice, you managed to get some debug messages. Well done.

This is the issue here; 'Ethernet shield was not found.' So the ardiuno isn't seeing the ethernet shield. I'll be honest, if you don't know much about arduino, I think this may be a little hard to solve. It looks like the issue is the arduino isn't seeing the ethernet shield. I wrote some code for arduino and the ethernet shield some time back ( here ), and it was a hard one to get going, and I spent a lot of time on it. I was using a clone Wiznet shield and had a hard time getting it to work. I eventually did, but only through trial and error. It's not really a setup I would recommend these days unless you know arduino pretty well. It seems there are some differences between ethernet shields.

To get this to work you are going to have to get your ethernet shield working properly with the code, and may require some trial and error changing pins in code and stuff like that. Personally, I would recommend, if it is possible for you, to get a Wemos D1 mini or good clone (ESP8266) and use that instead on wifi. There are a few examples of code in the arduino code forum. It is a lot less messing about as the network stuff is all integrated on the Wemos D1 mini so is really a case of connecting your counter via the three wires, grab some code and change your user/pass, wifi, and that's about it.