In order to receive Steam Trading Cards, you must play a game for some period of time. Is it possible to keep the game open as you go run errands, and have the cards when you get back, or do you have to be actively playing the game to receive cards?
Steam Trading cards are given out based on time spent in-game. Steam doesn't have, for most games, any way to know whether or not you're actually doing anything, just whether you have the game running or not.
It is also interesting to note that you can run many games at the same time to get the trading cards, and you can even get them from game launchers if you want to conserve computer power.
Yes. You almost don't even have to idle. I got mine pretty much just for starting up portal 2. It's pretty much the same system they have for item drops in TF2
Yes you can, but there is a set number of trading cards that will drop per game, unless you get a booster pack, which contains three cards for a game, and will drop randomly when other people craft badges.
For all free games, like TF2, you have to spend money in game to get trading cards for it.
Yes, you can. And since it's not mentioned in any of the other answers: there is even software that will iterate through your games, idling in each one until you get all the cards.
It's called Idle Master.