I have been with BT for my internet for 19 years. I know have Infinity and have regular download speeds of 38 mps, which is ample for my needs.
Don't forget, that if you access your ISP from your telephone line, BT is involved at the heart of things.
I am not very sure about your second question...there are plenty of techies on here that will help. But you can access your router from anywhere in the house, by wifi, which is what I do with my computers. I am not sure if you can do the same with your TV....perhaps you will need a cable for that.
I access my router with one of these :::
Amazon.co.uk User Recommendation
It works perfectly.