Is there a reason why you want to cancel your original card? Definitely hold onto it. If it's because it has an annual fee, call them up and see if you can downgrade to a card without the annual fee.
Can't check now but I think AMEX does allow you to transfer their points to Avios miles. I would wait for a promotion by AMEX since they usually give bonus miles like 30% that I remember seeing recently. Yes, you should be able to book through AA website using Avios miles.
Yes, you can also get the British Airways card by Chase. It's not as lucrative like earlier in the year but can be rewarding if you can meet the minimum spend for 100K points.