Alien Deleted Cocoon Scene
I personally thought they should have kept this scene in the movie. It is a very grim scene, because it shows that the Alien doesn't only kill people but wrap them up in this shit. I believe James Cameron used elements of this cocoon stage for his Aliens movie. Where I think here it is implied that the alien was going to implant the “seed”. Aliens shows us differently.
In the movie, Dallas is seen one last time with the Alien catching him in the ducts and that's all we see of him. In the theatrical cut, it is implied that he died. Here, we see that Ripley kills him out of pity.
From IMDB, ((After Ripley discovers the remains of Parker and Lambert, she makes another discovery. Ripley enters the landing gear area of the Nostromo (where Brett got killed) and discovers a cocooned Dallas and Brett mutating into an egg. Dallas pleads, "Kill me". Ripley flames Dallas and the Brett-egg and then runs to set the ship on self-destruct. ))
So, they were being turned into eggs. That's pretty gruesome. Given what we see in Prometheus, I could actually buy the different methods of the Aliens birthing cycle.
Planning Deleted Scene
This scene is a little rough, and clearly was meant to have cut away shots and closeups. I agree that this one needed to be cut.