AT&T likely won't allow him to buy his own modem, but a new router might help. Leo likes D-Link routers, so he could use that instead of the 2wire. But Leo thinks it's the actual AT&T U-Verse service that's the issue, though. It could be bad phone wiring in or around the house, so he'll want to call and complain to AT&T.
Yes. She is renting her router from them. She'll need two things to use her U-Verse connection. One of those things is the modem which takes the U-Verse that's coming in demodulates it into internet access. Then she needs a router which takes that internet access and allows a number of devices to use it. The unit she's renting is probably one box. Sometimes, it's possible to buy that box and eliminate the rental, but it's not possible with U-Verse.
When Leo went to China, he bought an international data plan from AT&T and his phone worked great everywhere. Leo doesn't recommend getting a SIM card in China. If Kevin wants to unlock the phone, he should ask Apple and AT&T. They will do it, but it's non trivial. Even though AT&T's plans may be expensive, there are a couple of reasons not to get a data plan in China: