Some songs on the soundtrack, including Let's Dance, Stayin' Alive (heard when Arthur and Selenia "dance" on a disc), and Theme From S'Express, post-date the 1960 setting. However, soundtracks are frequently separate from the movie's "world," so these songs could be purely for our benefit, not heard by the characters. Alternately, these could be deliberate anachronisms, included to induce a feeling of fantasy.
(at around 33 mins) Selenia is surprised Arthur knows her name when they first meet, but her father just named her to the whole audience. This is because when Arthur is being told the story of the Minimoy's by his grandmother, Daisy tells Arthur her name when he asks about it.
(at around 50 mins) When Arthur needs a rope and asks if Betameche would have one, Betameche says that he hasn't got any rope. In the next scene, when Selenia takes Betameche's Swiss army knife to pull down the poppy blossom, his knife has a rope that she uses for a grappling hook.
Minimoys size (2.5mm) is not consistent through the movie: bigger when in the straw, smaller when Arthur runs under an ant, much bigger when dancing on the vinyl disc, bigger on the model car, bigger on the ruby pile etc.
The house is specified in the movie as located in Conneticut USA, is actually a real house somewhere in the British Isles (England) because the electricity power points are United Kingdom 240 volt earthed type with switches, visible on the wall just to the right of the piano when Arthur blows out the candles on his birthday cake and visible on the wall to the left of the broken door on the floor in Granny's bedroom.
When Arthur, Selenia and Betameche are inside a straw segment that is floating in the water, there is no water inside the straw even though surface tension would not be enough to keep the straw from partially sinking into the water.
The story is based on the premises that a doorway into the Minimoys world opens up every 10 months, and the return is possible within 36 hours, or, if you miss that window, after 1000 days, which is almost 3 years (much more than 10 months). While it may appear as a contradiction, it really isn't. The inbound doorway opens due to the full moon, but the outbound doorway doesn't require the moon at all, since it can open even during the daylight. Thus, it's perfectly reasonable for each way to have different rules.