Personally I found running many games at once does not work for me, I have gotten cards when it was 3 or less games at once.
Cards generally drop once every 30 minutes for most people when playing a single game I am unsure if this average stays the same when using multiple games.
I do not believe there is a limit to how many you can obtain in a day since there is a limit already associated to the total amount you can have.
Things to keep in mind
Level 10 you start getting foil cards.
Every 10 levels you increase your chances for booster pack by 20% at 100% it has (doubled)
Sources