Can there be trading cards in free games without paying anything?

Solution 1:

Different games have different requirements. Publishers/developers can choose whether the card drops will drop through in-game purchases or through buying a DLC. In the case of TF2, they choose the in-game purchases option, that means you can only get your card drops after spending $9 in that game.

Steam counts activating a license that is free as purchasing it, regardless of if you are paying something or not. So a free game like Grimm works, because you are "buying" a DLC for that game (even though the DLC is free), so you now are able to get the card drops for the base game. You do have to activate the DLC, without it the cards don't drop.

For a game like Relic Hunters Zero, it will only drops cards if you got a key from somewhere and activate it as opposed to getting it for free through steam's store.