So, this is really three questions, but they're kinda related to "how do talk show appearances work", so I'll try to answer them all. (You probably should split these up into multiple questions and I can split my answer accordingly.)
First, the bit with the $1,000 was almost certainly a gimmick, and Ford probably gave the money back before he left. There was a similar recurring bit on The Late Late Show with Craig Ferguson where guests could "win" money, but it was entirely for show. I'm not positive but I think there might be some rules around giving guests money for their appearances; see #3.
Second, you're a bit confused how American television censorship works. There are two different aspects to American TV production where censoring gets involved: the government and the network.
There is a government agency, the FCC, who's actual job is to manage the use of the electromagnetic spectrum for radio broadcasts -- which includes over-the-air radio, television, cell phones, short-wave, etc. They parcel out chunks of bandwidth in various parts of the spectrum to licensed operators. In order to get such a license, you have to agree to their terms, one of which is that you won't use the public airwaves to say naughty things. If you do, they fine you lots of money, and potentially pull your license. The key here, though, is that the FCC only has jurisdiction over broadcast media. This means that anything intended for broadcast by the "big 5" networks -- NBC, ABC, CBS, Fox, CW -- or by local over-the-air broadcasters has to adhere to FCC rules.
Satellite and Cable TV, which is not "broadcast" but "directed" transmission (in other words, you can't pick up cable/satellite transmissions "by accident" with just an antenna) don't technically fall under FCC jurisdiction. However, the networks themselves also worry about the public backlash -- especially advertiser backlash -- if they allow too much inappropriate content to make it out on their channel. So they also have internal rules over what is and isn't allowed, but generally these vary a lot from one network to another. In the case of Conan, TBS is not available over-the-air, only on cable/satellite, so the only rules he needs to adhere to are the TBS networks. Since the show airs very late at night, they're not quite as worried about the occasional spurious expletive going out. (There was a South Park episode that aired on Comedy Central who's entire premise was saying "shit" 162 times.)
Third, you're correct that the guests on these shows aren't typically paid by the show to come on. The show pays all their expenses (hotel, flight, food, etc) but they don't pay the guests for their time. After all, the guest is usually on the show promoting something, so there's a mutual benefit involved.
Technically, if the guest happens to be a member of SAG or AFTRA, they're probably under a collective bargaining agreement with the studio to pay them for their time; my understanding is that with most actors, their appearances are part of the "promotional tour" before a big movie, and is covered as part of their movie contract. Thus, the movie studio would pay them a set amount and they're contractually obligated to appear on a certain number of programs.
If, for some reason, the actor isn't being compensated by the movie studio, then the show would be obligated to pay them -- though the amount is tiny compared to what they make (it's the union base rate, which is less than $1000 per appearance). Also, that stipend would only apply to actors; other guests (authors, politicians, musicians, etc) would each have to be treated according to whatever agreements the TV studio has with them or their unions; in any case, they're not getting paid for "a days work".
In certain cases, a show might want to lure a guest on that would not otherwise appear, e.g. someone newsworthy they want to have an "exclusive interview" with. This seems pretty rare as far as late night talk shows go, but you see it a lot during daytime programming. Legitimate news programs usually have rules against paying for appearances, but "news magazine" programs may not. It those cases, the usual contract negotiations would happen, and the show's production company would have to compensate that person however much it took to get them on the show.
(Thanks to @JakeGould for finding this quora article; it's not specifically about Conan but the same rules generally apply to most talk shows.)