Trading Cards; how they work and stuff?

You need to purchase something from the Dota 2 store to get cards.

For the Free to Play games Team Fortress 2 and Dota 2 you need to buy an in-game item in order to get cards to drop.
For your other question in the case duplicate cards that does still count as a card drop and it will not drop any more cards while you play that game. HOWEVER you can randomly get a booster pack for any game you have gotten all the drops for as long as you have logged into steam within a week. They are kind of rare though. Otherwise yes, you'll have to buy or trade for the rest of the set.

As you already pointed you get HALF of each game set. DotA2 and Tf2 and Warframe required (to get card drops) to pay in-game money to earn more drops. And when a game says "you have 0 drops left" means you'll never get any cards again from the game BUT there's a thing called booster pack, which allows you to get 3 cards for a random game YOU OWN and that you have 0 cards left IN IT. Booster packs drop randomly after the community crafts enough badges.