Brian decided after 3 weeks, on April 21, that he had to see her again. That means it was March when he met her. Neither was dressed appropriately. She walked with Brian while wearing a sun dress after the museum. Trees were all in bloom. The weather in NY in late March is often FREEZING but OK let's assume it was warm. But the trees could not be in full bloom.
When Brian and Arielle go to the Guggenheim, they view Edward Hopper's Nighthawks. This painting is not in the Guggenheim, however it could be on loan to them (which seems very likely, considering The Long Leg is not in the Guggenheim, either).
When Arielle shows herself wearing Brian's ring at the end of the movie, it isn't clear how she re-obtained the ring. Earlier in the movie, she returned the ring with her goodbye letter. However, Brian subsequently puts the ring back in the empty envelope and hands it to the doorman, asking that he returns it to Arielle.
From about 10:20, as Arielle and Brian walk down the path, a couple of people turn to stare at them. Considering that a couple walking down a path isn't anything worth ogling at, it is obvious that they got a sense that this couple was being filmed.