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Yes, you can have multiple copies of a card even after you used that type of card to craft a badge.

I'm pretty sure it's doable, as people with level 5 or foil badges still can get booster drops ( = cards) for the games. Having level 5 badge won't close down your inventory for those cards :)

You can have as much cards as you want.

There is absolutely no limit, as Lugum said. I have seen large-scale card-trading bot accounts hold more than 100 levels of worth of card sets for the same game (CS:GO).