I tend to agree with you that the best thing to have done was to send the guests home and hold a quiet ceremony, and perhaps they did that; nothing is ever stated about the wedding. But since J.D. and Tom are on a first-name basis (as we see near the end: the rich know the other rich) it's logical to assume that most of the guests either knew, or at least knew of, Brookman, so having them stay on might not be so implausible.
Regardless, the scene at the diner was definitely that same evening (remember Schatze's latest tiresome and gross crack about how "Following the ceremony, the guests adjourned to an exclusive greasy spoon where perfectly delicious dogburgers were served"?), so it makes sense that she and Tom were wed at her apartment while the judge, guests and all the paraphernalia were there.
Or -- to argue the contrary -- maybe not: wouldn't many people at the party have known who Brookman was and have inadvertently spilled the beans? In that case they might have held a private ceremony after having the guests depart. (The guests would have to have been fed and given some courtesies for attending a canceled wedding.) Even so, it would likely have been in the apartment using the same judge and having at least a few people present. As to best man, either Eben or Freddie could have easily stood in -- my guess would be Eben, since he's a pretty nice, straightforward guy.
Anyway, specifics aside, the pair were plainly married that very day and dined at the burger joint later that evening.
Atlantic City is a town on the southern coast of New Jersey, about 125 miles from New York. It was developed as a resort beginning in the 1880s and 90s and reached its peak in the 1920s and 30s, known for its famous Boardwalk and many tourist attractions (and, during Prohibition, for bootlegging and the ready availability of liquor). But after the war it declined as people could afford vacations in other places, including warm weather spots like Florida or California, year-round. By the early 50s it was beginning to slide downhill and reached its nadir in the 60s and 70s, riddled with crime, abandoned property and racial problems, before the state passed legislation legalizing casino gambling in the city in 1977, after which the place boomed for 25 or 30 years. But as casino gambling has spread and traveling to better climates and more attractive destinations become ever easier, AC has resumed its decline, with many casinos (including Donald Trump's) going under in the last couple of years and the area falling into an economic slump once again.
Truth be told, the place was always something of a dump, with a thin veneer of glitz that couldn't conceal its crummy aspects. A good movie to see, made during the transition to the casino industry is, appropriately, Atlantic City (1980: originally titled Atlantic City, USA, since it's a Canadian-French co-production), with Burt Lancaster and Susan Sarandon (both Oscar-nominated) and directed by Louis Malle, which gives you a good idea of what the old place was like. But the new prosperity promised in the film proved shallow and fleeting too, as lifestyles changed across the country over the ensuing 30 years.
Anyway, Schatze wasn't far off the mark when she said, "Nobody's mother lives in Atlantic City." Yes, obviously, there are many mothers in the city, but it's not a place where someone rich comes from or still lives in, and the line itself should have put Pola on guard; nobody goes there except for a day or two of hanky-panky. And how a guy with a strong French accent and "foreign" demeanor could have hailed from AC, or had a parent there, is really part of the joke. I think it's meant to convey how gullible Pola is. Oh, by the way, no one flies from NY to AC; while it's possible to do so (there is a small airport), then as now most people get there by train or car -- mostly car. Also, a prop flight to AC from New York's LaGuardia Airport (which is where they showed the plane taking off from) would take less than an hour. To Kansas City, Missouri -- right in the middle of the country -- by prop would have taken about six hours in 1953. You'd think even Pola would have wondered after the second or third hour why the plane hadn't landed.
Addendum: By the way, here's the link to the Wikipedia page on Atlantic City:
It mentions the fact that many of the city's street names were used for the board game Monopoly, and it also has a small section on the HBO series Boardwalk Empire, which ran from 2010-2014 and was an excellent, if somewhat fictionalized, depiction of the boss system and its crime component during Prohibition. It starred Steve Buscemi and won many awards. I liked it very much.