After the group ties up the dead soldier to take to King's Landing, Jon Snow tells Gendry to run back to Castle Black and send a raven to the queen for help, Tormund tells Gendry to give him his hammer because it will slow him down, and after they start running up the hill you clearly see Tormund holding his battle axe in one hand and Gendry's hammer in the other, but then the very next shot the hammer is missing.
It is established that it's cold enough to refreeze the lake in a short time (days or hours). Jon Snow pulls himself from the lake and his clothes have no way of drying off before they freeze on his body. In temperatures a few degrees below freezing, this would not take very long; the wind would have frozen his clothes stiff even faster.
When Jon Snow and Tormund first spot the small band of wights following a path below their snowy perch, as Jon approaches the edge, there is an obvious hand print already waiting in what should be fresh snow on the rock where Jon places his hand to lean forward and look down to see them.
While Jon's squad is battling the wights on the frozen lake, next to Beric, you can clearly see a man swinging his weapon at nothing. The animators obviously forgot to add a CGI wight as his opponent.
There is no movement whatever when Jon and Daenerys are in his cabin on the boat. Ships are not that stable, even on a calm sea.
It is not explained why Jon and his crew don't simply break the ice around them as soon as it forms to ensure the dead are never able to reach them.