A few things about your deck:
A) Your deck includes all the available cards for a character of your stats, plus the cards relevant to your current storylines (Making your name, living stories, etc). As you progress, some cards are left behind, and other, more profitable ones, are added to your deck.
B) There is no need to hold on to the Gift card for half a year - it’s not a rare card. If you discard it, you will soon draw it again.
C) some cards are available to everyone, some depend on your progress along a story. There are some location-specific cards, too: almost every location save the Forgotten Quarter and the Labyrinth has one or two of those.
D) Faction and pet cards are available everywhere.
E) The six (or ten) cards in your deck are not pre-selected. Every time you click on the deck, the cards are randomly selected. You may see the same card appear three times in a row.