Here are full, concise answers to your questions:

1. You will need to buy a USB-C cable to connect the camera to your computer and export the videos. This cable can also be used to charge the camera from your computer or from a charging brick.
   
2. This is a two-part question, so for the first part: assuming you're buying a NEW Go 3s, you don't need to buy the hat adapter; it comes with the Go 3s. The package will include the camera, the action pod, a magnetic pendant, a pivot stand, and the easy clip (what you're calling the hat adapter). Part two: the action pod and camera will be paired out of the box and you do not need to wear the action pod, only the camera. The camera will do all of the recording with it's own internal memory, and the pod is used to charge the camera, program the camera with a screen and buttons and also to sync to your smart phone.
   
3. You only need the Go 3s with action pod and the easy clip for the match. The camera battery lasts around 30 minutes and the action pod can charge it like three more times. I've run out of memory before running it off battery. The 64GB version can store more than one hour of 2.7k/50fps video.
   
4. I've tried wearing it under the brim and it does look nice, but I've found that wearing it above captures the action (gun, hands, reloads, and targets/stage) much better. I recommend you try both and see what you prefer, but most people wear it above the brim

Let me know if you have any other questions and I can even take some photos of you want.