When Arthur is explaining to Takumi that he is going to get help and bring them back to rescue him, it changes from Arthur with his hand on Takumi's face in the next shot to Takumi having his hand up to Arthur's face (and Arthur is no longer touching Takumi at all).
When Arthur is shown on the airplane, he has the orchid with him and then again when he arrives home (he is shown setting the potted flower down). An agricultural item such as this would have to go through quarantine and he would never just take it home.
When Arthur comes home to find Joan asleep on the sofa, the tea kettle starts to whistle. He comments that she could have burned down the house but this is in error as 1) she had only fallen asleep in the time between turning the fire on under the kettle and 2) she awoke to the kettle's whistle.
Taxi drivers in Japan always open and close doors, not passengers. It's a safety measure and the controls are operated from the driver's seat.
Whether orchids appear in Aokigahara when a soul passes into the great beyond may be a bit of directorial license but one definitely can't bring orchids or any plant with soil back into the US.
The terrain up hill from the freak flash flood was to short to accumulate that much water. Moreover there were no signs of previous erosion, apart from the 'cavern.' Finally, the characters would have been pulverized if they had been in a torrent with rolling boulders of that magnitude. Just the director's way of adding a bit of drama to an otherwise uneventful plot.