If the state gives money from your taxes to you, or waves that amount for you in return for what you gave of zakah or charity to charitable organisations, then the money that comes back is yours, and it does not affect your zakah.

The husband does not have to pay Zakah on the jewelry on his wife’s behalf, because the Zakah is only obligatory on the possessor of the wealth, and the possessor in this case is your wife, because she is the owner of the jewelry.