I want to convert this GMT time stamp to GMT+13:
2011-10-06 03:35:05
I have tried about 100 different combinations of DateFormat, TimeZone,
As always, I recommend reading this article about date and time in Java so that you understand it.
The basic idea is that 'under the hood' everything is done in UTC milliseconds since the epoch. This means it is easiest if you operate without using time zones at all, with the exception of String formatting for the user.
Therefore I would skip most of the steps you have suggested.
Alternatively, you can use Joda time. I have heard it is a much more intuitive datetime API.