You all overlooked the main story. Explanation here.

First of all sorry for my english.
I am surprised that most of people overlooked maybe the main story of the movie. Dealing with death of the partner and finding your own voice. Cat represents Mike.
In first scene he puts on old record which he recorded with Mike. Cat gets out and he must carry it around. He is carrying around cat/Mike all the time whenever he goes. He lost the cat and bring the other one. His emotions to Mike are not in the right place. He didnt deal with his loss. Bud Grossman said to him that he must get together with a partner. He was not a solo act at that point. He still had his partner in his mind. While driving from Bud he saw the bridge where Mike killed himself and then ran over the cat. He had a feeling that he hurt memory of Mike by trying to successed alone. When he gets to Mike's parents house cat is there. Llewyn gave up on music career so his memory of Mike is again there unintact. They told him the name of the cat. At this moment he lets go. The memory of him belongs in Mike's parents house. At the end he doesnt let cat/Mike out of apartment and sing a song that Mike wrote. He doesnt carry him around anymore. He finally accept it. He is a solo act.
Coens are trying to tell that acting one way will get you the same results in life. Thats way there all seem like a loop. But when you learn something this "loop" can get a little different everytime. In this loop he resolves loss of Mike. He still have issues like being a jerk and not connecting with people thinking he is more than others or sleeping with girls that he shouldnt. That explains second pregnancy. That "loop" happened before and he didnt learn enough. Thats why he repeat it. This time around he learned that previous women didnt do a abortion. Now he must know that Jane probably wouldnt do a abortion. This things will be resolved in next "loops".

Story is very complex and its dealing with alot of stuff. I saw movie only once so I am sure I miss things and maybe not explain all cat/mike story right but I am pretty sure that cat represents Llewyn emotions towards Mike. I think I explain meaning of a "loop" right too..