Websites suddenly inaccessible but Google search works
At the moment a large number of popular website are dropping support for all SSL/TLS versions below TLS1.2 because of the known vulnerabilities in TLS1.1/TLS1.0 and older.

Your device is very very old and does not support TLS1.2 therefore you can't connect to certain sites like https://wikipedia.org which requires TLS 1.2 according to the test of SSLLabs.

If you use extreme/compressing mode of Opera Mini you connect to a proxy at Opera instead that compresses and re-encrypts the traffic. Therefore using this redirection you can still access those sites as long as Opera not also increases the minimum TLS version on this proxy.

Some Webbrowsers may also include their own TLS implementation with TLS1.2 support. But usually they have dropped support for Android 2.x long time ago.

Conclusion: Your device finally comes to an end. If there is no newer custom ROM available (AFAIR for TLS1.2 Android 4.2/4.4 is required) allow it to rest in peace and get a new phone.