When Adeline dives into the water, she is diving away from Jean and towards the sun. Yet when the shot is on Jean, watching Adeline, the sun is behind Jean. In the next shot, the sun is clearly on Jeans left shoulder and as Thomas wakes and comes on deck and speaks to her, the sun is now on Jeans right shoulder.
The state known as Germany was unified as a modern nation-state in 1871, when the German Empire was forged from 39 sovereign states, Prussia being the largest. While the exact date that Louis Wagner arrives on the Island is unclear, it's unlikely that a Norwegian immigrant such as Maren would have assumed that Louis was "from Germany", when he could have been from any of the 39.
When Josh Lucas dives into the water of the cove, it is almost perfectly still and glass-like. As they show Jean photographing him (in black and white) at the exact same time, the water is very choppy indicating there is a stiff wind blowing.
When the tea kettle reaches boiling, we see steam escaping not only from the lid but also from the completely uncapped spout. And yet there is a whistling noise. Steam doesn't just whistle. It has to travel through a pressure release valve to make a sound. This kettle clearly did not have one.