An unlocked phone is one that can be use with any carrier (that supports the phone technology GSM, CDMA, etc). If you got you BB used, and it was on a BES it may well still have an IT Policy loaded. Options -> Security Options -> General Settings have something about the IT Policy if there is one on the device.
The Blackberry web browser is designed to work through the Rim Operations Centre using the communications support of a Blackberry Data plan, so a third party browser may be your only choice. A blackberry is not just a regular phone, there are reasons why there are Blackberry Data Plans. That said, there should be no reason for Opera won't work using the APN if you have it set up correctly. You don't need the username/password in the APN setting. It has been a while since I used Opera, but IIRC there are some settings that let you select whether it uses BES, BIS or direct TCP.