ISP Provided Router vs. Personally Purchased

I'm tempted to try and convince my mom to buy a new router rather than using the one provided by our ISP and have a few questions.

1. Is it true some ISP's backdoor their routers?
2. Can I boost our speed by getting one?
3. When accessing the router settings using the gateway will I be able to be able to block connection to sites/ip's if I'm currently unable to do so? (Specifically to mitigate ban risk on my cfw ps3 when online it recommended to block a few specific web addresses)
4. Does AT&T (our ISP) charge a rental fee for usage of their supplied router?