How to make new backgrounds is in the Dungeon Master's Guide in the “Creating New Character Options” section that starts on page 285. How to make new backgrounds is on page 289.

If the players will be creating the new backgrounds' concepts, you should work with your players on the mechanical details, not let them choose them on their own. Creating new backgrounds is potentially quite powerful in the hands of a self-interested player, and they shouldn't be let loose like kids in a candy shop. Let them do Steps 1 and 2 in consultation with you to determine theme and flavour, and then you do Steps 3 through 5 yourself to determine mechanical benefits, and then consult with them to see if the result suits the theme.

Your confusion seems to originate in how you create characters that are above level 1 when starting. The players' handbook doesn't have a way to create characters that start above level 1, other than creating a level 1 character, and then leveling him up to the appropriate level. Let's take the level 3 bard as an example.

When you create your level 3 bard, you start out with building a level 1 bard. He has 4 level 1 spells, chosen from the bard spell list. When he reaches level 2, he gains another spell known, and can exchange 1 known spell for another one. He now knows 5 level 1 spells. When he reaches level 3, he gains another known spell, along with 2nd level spell slots. He can also exchange one known spell for another one. He now knows 6 spells, and up to 2 of them can be second level.

For your character generator, I recommend either only doing first level characters, or doing first level characters you can then level up. This is most intuitive for you to program, and doesn't overwhelm the users with too many options immediately.