It's normal to change the status to "sold" and rightmove do stipulate on their website that it's "sold subject to contract". Once it's actually completed it will come off the website altogether and will be no listing on there. If your offer falls through they'd just remove the sold status so people could see it on the market again.
If people want to they can choose to filter their searches to include sold stc properties but most people wouldn't bother. You can continue to show other people around the house if you want to, however if the current buyers get wind of it they could pull out or be less receptive to negotiations. We pulled out of an offer after the buyer was continuing to try to get a better offer - we didn't trust he was going to go through with the deal and we weren't prepared to start investing money in solicitors and surveys if he was just going to mess us around. If you've had a serious offer and accepted then you'd be being unfair and possibly foolish to keep the house on the market.
Has the buyer got a mortgage approved in principle? Normally a buyer gets a mortgage approved in principle for x amount so they know they can go ahead and make offers, then once an offer has been accepted the bank do a valuation (survey) so satisfy themselves that what they are lending is no more than the value of the house, then they'll finally approve the mortgage. If this is the stage your buyer is at then great, if they've made an offer without having a mortgage approved in principle I'd be a little nervous but wouldn't be looking to back out. However if the process started to drag out because they weren't yet approved for the money you could start then throwing in conditions like they need to get mortgage progessed by x date or you'll put it back on the market.