We have put together a short survey to capture your feedback in terms of what BT can do when you experience home Wi-Fi issues. We would appreciate it if you could take a couple of minutes to complete a short survey. Read more from here, Fix my home Wi-Fi
You would need to have your own line installed into the house and sign up for a possible 18 to 24 month contract for Broadband. Suspect not cheap for a 16 year old
The cheapest package is just under £30 per month on a 24 month contract. You will need to pass a credit check and no doubt because you will not have a credit history, pay some sort of deposit. You will also need to get your fathers permission to install a second phone line on his property.
Perhaps you just need to suck it up that you live in your father's house, your father pays the bills, it's his broadband and he can do what he wants with it and that includes deciding when he wants to turn it off.