It might seem contradictory that the aliens have trouble getting past doors that have only a single board over them when they can jump onto rooftops, but in real life, jumping ability depends on one's strength to weight ratio. An elephant can't jump, but it is much stronger than a cat, which can jump insanely high. It's entirely possible that the aliens are not incredibly strong physically, but they are lightweight.
The aliens have biological camouflaging abilities, but their camouflage is imperfect. That is why they are mostly active at night.
The crop circles are used as markings for the aliens. Therefore, they are 'signs' for the benefit of the aliens.
Why would the aliens use crop circles, which were previously revealed to be a hoax, for their invasion plans? Because then they can set them up in a way that discredits anyone who believes they are linked to aliens.
In several scenes when the aliens are near, the wind is blowing, which rattles the wind chimes on the farm. Given that even their ships can become invisible, it's possible that the wind is caused by a nearby alien ship touching down somewhere near the house... which might mean the aliens were right there when Merrill was staring out into the cornfield just before the invasion took place that evening.
Many, many people have questioned why aliens would invade a planet covered in a substance fatal to them, or at least do so while not wearing any sort of protective gear.
