I've had one occasion where my CC number was cloned. Somebody ordered a bunch of crap and I caught it on the statement. Called them up and they took care of it right away -- froze the card and sent out another one, and I wasn't liable.
I've occasionally had a problem with goods not shipped. Got them resolved right away.
Any credit company will take care of you. Their entire industry is based on trust, and you the customer probably have a dozen offers from competitors every week in the mailbox.
I'd suggest that you involve your credit card company right away. Just let them know that you're having a dispute with the furniture vendor, and what steps you've pursued with them. The agent you talk to at the CC company has experience in this matter... and they'll have better advice than anything you'll get here. Just call them, fill them in, let them know that you have your evidence trail in email. If they want to take over, let them. Even if you think you've got things in hand with the vendor, the earlier you get the CC guys involved, the easier for them to do their job when/if they need to.