You should schedule dining for any of the days of your trip where you want a table service experience, especially in any cases of character dining, signature dining, or dinner shows. This is regardless of whether or not you have park tickets (park tickets are not required to make dining reservations, though park admission is required in order to dine at any restaurants within the parks). You can begin making dining reservations 180 days before your trip (180 days + 10 if you are staying on site at a Disney resort). Counter or quick service does not take or need reservations (the exception to this is Be Our Guest).
There is no significance to Magic Band colors - they are chosen based off your personal preference. If you don't choose to customize, I believe the default color is grey.
There is no character dining with Toy Story characters, but Woody and Buzz do a character meet and greet in Disney Hollywood Studios. There are also often the green army men characters in that area who will stop for pictures.