During the scene, they cut to a close up of Ripley's face several times. In some shots, you can see the glass of the helmet she's wearing, in other shots, her face is uncovered. The alien face hugger sacs were apparently suspended and filmed from below, then the film inverted to give the appearance that the sacs were rising up from the floor. In the very next moment, Lambert has it in her hand. It would not seem likely that Lambert picked it up while everything near it was burning.
Every ship nowadays has enough lifeboat capacity for everyone.
After the alien leaves Kane's face and the rest of the crew are looking for it in the infirmary, there is one shot that bears re-watching. When Ripley is looking upwards over the prone Kane, you begin to see the dangling tentacles of the alien in the background. At the same time, the collar on her outfit begins to pull downward, un-naturally, until the alien drops onto her shoulder. I believe there was some sort of string or rope fed from below, through her clothing, and was attached to the alien and was pulled to have the creature drop onto her. There is a scene when Ash is beating Ripley after she confronts him about protecting the alien.