When Stingo is packing to go back home and Sophie comes back home and is talking to him, you can see a pair of Nike speakers on the bed. Nike was established in 1964, much after the time when the movie is set.
Slavic surnames ending in -ski/-skiy are, in Slavic grammar, considered adjectives, and so the female form is -ska. Sophie's and Eva's surname should therefore be Zawistowska. Moreover, it is unlikely for Eva's name to be spelt with a 'v' as the proper Polish form is 'Ewa' ('v' does not appear in the Polish alphabet and is only used in foreign names and loanwords).
When Nathan and Sophie drink what Nathan calls a Chateau Margaux, the color of the wine in the glasses looks nothing like a Bordeaux, let alone a Margaux. It looks like a cheap rosé wine or a very badly made red.
When Nathan toasts Stingo on the bridge, as the shot pulls back you see first one then several cars from 1982, obviously an issue with how long they could hold up the bridge traffic for the shot, they needed to hold it up an extra minute or maybe lacked the budget to pay for late 40's cars to drive by instead. You can also see the lights of The River Cafe, a waterfront restaurant under the bridge that didn't open until 1977.
In one of the early scenes when Stingo is moving in he is carrying 3 cases of Spam on his shoulder. They barely move despite him writhing around to get the door open. When he gets into the room he drops them on the bed and you can clearly see most of the cans are glued to the cardboard. The actor even flips the top row over on the bed and they stay attached.