If a new customer (US citizen w. SSN) wants to open the account in the states. But he has a resident address in south america. the customer provided the C/O address in TX.
In this case, should we have the address verification for either South America resident address or US mailing address?

There is another customer. He wants to have a different name appearing on the account(from his legal name). We documented our internal data with his legal name. but could we still have a different name on his account, when we consider AML rule?

The CIP regulatory requirement is that the customer provide a residential or business street address. It doesn't require that it be the customer's primary residence. It does permit the use of a "contact" address if the customer has no residential or business street address of his/her own.

In this case, unless the customer resides in the U.S., you have to use the South American address for CIP purposes. You can still use the U.S. address for mailing.

As for the customer who wishes to use a different name on his account, that's a thornier question. There's nothing inherently illegal about using an assumed name on an account if there is no intent to defraud. However, my sense is that most financial institutions require the use of a name that can be documented as recognized by a government (ID-issuing) agency. This is a question on which your institution should have a policy, and you should adhere to it.