Personally, I wouldn't use Opera Mini for banking or similar because, whilst the data is encrypted between your phone / Opera and between Opera / the website; it has to be decrypted and processed on Opera's servers. There is only a tiny tiny chance of any security risks, but as they sometimes say Don't use Opera Mini for anything you don't want Opera to see.
Having said that, their servers are probably more secure than our home computers.
A proxy is a computer system that carries your request rather than it going direct to its target. To the target (eg O2 Active ) it seems as if the request is coming from the location of the server rather than your own location. The proxy can forward your own address to the target so that it knows who / where you are and that you are using a proxy.
This is how sites like Google know that you are in the UK, even though Opera's servers are in Norway, Poland and Iceland.
A transcoding proxy is one that converts data that passes through it to another form. In this case from a webpage (HTML) to an Opera Mini page (OBML).
The reason that O2 can't identify you is that there are more mobile customers than IP addresses, and so mobile networks share the addresses outside their networks. So O2 can see that the request us coming from Opera and that it was originally from its own network, but only from a range of customers.
