If you have a signed contract then it should be marked CTG (contingent) or PEND (pending) in the main MLS (multiple listing service or the likes of). This lets agents know that their is a contract on the house. This will usually stay this way until closing and then once the house closes it will get closed out in the system.
The sites that pull from the main system are sometimes behind what the actual status of the home is. House can show up as still on the market even after they have been sold and closed.
If you have a signed offer and are past the contingencies any offer that would come in would now be a back up offer. You offer would have to be canceled for the next offer to move forward. Hope that helps.
In my location (Silicon Valley), when we say off market, we mean not advertised other than by word of mouth through the Broker/Agent network.
Also, a GFD (which I think you're calling a Good Faith Deposit) is more commonly known as an Earnest Money Deposit (EMD) and is NOT anything other than the beginning (and sometime all of) your down payment. It's just provided up front so the seller has some degree of assurance that you're serious about the purchase. A GFD/EMD is not an additional amount or any kind of compensation for the inconvenience of removing a home from the active market. I sincerely hope a Realtor didn't tell you that.
The bottom line is this. Once your Offer is accepted, you submit your GFD/EMD to the escrow holder. Having a purchase agreement that has been accepted by both buyer and seller is all the protection you need. Neither the buyer nor seller can back out of the agreement unless the other side fails to perform or one of the agreement contingencies are not met. The EMD is what the buyer risks by backing out of an agreement for no good reason. The seller only risks the time he's not active on the market.
Also, most MLS systems give brokers the option to place a home in a "pending sale" status. Pending sale usually has several options, including "Pending, Continue to show" which I use if a buyer for my listing has signed a purchase agreement with my buyer but they have not yet removed their contract contingencies. Agents can still show the home with this status, and as a listing agent I'm likely trying to collect back-up offers during this time (in case the first agreement falls apart). "Pending, No Show" is what I use when those contingencies get removed. At that time, the property stops getting shown and we're usually close to closing the transaction. "Pending Release" is another form of Pending that indicates the property is under purchase contract, but the contract contingencies are not yet removed.
Sorry, I got long winded. I hope this helps some:)