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Yes. Miracle triggers upon revealing the card when you draw it. You cannot cast the card until the triggered ability resolves and you pay the cost. Until that time, the card is in your hand and subject to the effects of discard.
It depends on whether the Miracle card is a Sorcery or Instant
It would be correct to say: Your opponent can make you discard a Miracle card before you can cast it for its Miracle cost, as user1873 already pointed out.
If the card is a sorcery, you can't usually play it outside the main phase or with a non-empty stack, so for your specific example, he could make you discard before you can cast it in any way.
If it's an Instant, however, you can still play it for its regular cost in response to the Funeral Charm.