Roger and Brianna cut Jemmy's hair because he has lice. That would not solve the lice problem. Showrunners did it to show hereditary trait he received from Roger to prove he was biological dad but that wouldn't solve lice issue.
It is apparently several months since Brianna discovered she was pregnant, since it has turned from winter into either spring or summer. But during the sex scene with Roger, Brianna doesn't appear to be pregnant.
Brianna puts her head to her son's head, being fully aware that he has lice and that this immediately transfer them to herself as well.
Claire's and Brianna's legs are shaved even though safety razors don't yet exist, women didn't shave their legs then, and they always wore long skirts down to their ankles.
The fact that Jemmy and Roger share the same birthmark is not conclusive evidence of Roger's paternity until it's established that Bonnet did not also have the same birthmark.