I've heard that if my trainer ID is a specific pattern or has certain numbers, it will effect how often I encounter shiny Pokemon. Is this statement true and what kind of patterns/numbers are most effective? I know the trainer ID is random at every new game, but is there a way to manipulate it without using any type of cheating device? For example, does my trainer name or gender selection determine my ID?
Your Trainer ID and Secret ID is generated by the seed that is randomly generated when you start the game for the very first time. It is not defined by your name or gender. There is no possible way to change your IDs without restarting completely or using a save editor or cheating device.
In Generations III, VI and V, your Trainer ID and Secret ID is used in conujunction with a Pokemon's personality to determine shininess. There is a 1/8192 chance of getting a shiny Pokemon.
First, your Trainer ID and Secret ID are XOR-ed (Exclusive OR - one or the other but not both, Ex. 11010111 xor 01101010 = 10111101). That result is saved as variable E.
Then, the encountered Pokemon's two personality values are XOR-ed, with the result saved to variable F.
Finally, E and F are XOR-ed one last time, then checked to see if the final result is less than 8. If so, your encountered Pokemon is shiny.
Here is an example on Bulbapedia, which includes all integer to binary translations.