If you catch a Pokémon does your chances of meeting it in the wild increase

The gender ratio for the 6th generation starters is currently unknown, but all starters so far had a ratio of 12.5% female / 87.5% male, or 1 : 7.

The gender of your Pokémon does not depend on your character's gender. Both male and female protagonists can have either male or female starters.

This specific gender ratio is used to make it more difficult in breeding more of them since players usually get them as gifts or Fossils rather than catch them in the wild. All starter and Fossil Pokémon are in this group.

The only way to ensure you'll get a female starter is indeed by saving the game beforehand and reloading it as necessary.

The chance of getting a shiny has indeed improved. It has doubled. This is further increased by chain fishing, getting multiple Pokemon in a row without moving or getting the "Not even a nibble" message. Poke radar works much the same to improve the odds, and the Masuda Method of breeding further increases these odds.

The Masuda Method means you take a foreign Pokemon and breed it with one from your region. For example, say you wonder trade and get a Japanese Pikachu. Breed that with a Pikachu you caught and you have a better chance of a shiny than if you breed two Pikachu that you caught.

This chance is increased further in the Friend Safari. In the Friend Safari, you are eight times more likely to find a shiny than the rest of the game. However, the shiny charm does not affect this.