If a lawyer is paid $10,000 to handle a case, then the lawyer later withdraws, is the client entitled to any refund?

It can depend on what the retainer agreement provides, but generally the client is entitled to a refund of theunused portion of the retainer, if any.  If there wasn't any set hourly rate, a Court would likely calculate the reasonable value of the work that was done (plus disbursements) and, if less than $10,000, award a refund of the difference.