This might be a stupid question, but I wonder where Michael lives after he comes back from WWII. Kay lives in Manhatten, I guess, so is Sonny and his familiy. What's about Fredo?

I believe Kay was still living with her parents in Vermont (New Hampshire?).

Sonny and his family lived on the Corleone compound. I suspect that Fredo lived in his parents' house. (It was Connie and Carlo who lived in Manhattan).

Are you sure about that? There is this scene where Michael an Kay have dinner at Kay's place. Is this her parent's house?

There is a scene where Clemenza visits Sonny at home after the assault on Vito. Is this Vito's house. Or is there another house on the Corleone compound where Sonny lived?

Again, where did Michael live after her came back from war?

If you read the book, Sonny lived in a house on the compound. There were several houses. I thought I read somewhere in the book that Michael had an apartment somewhere in the city, but I can't seem to find it at this time. I'll have to research that further and let you know later.

Thank you, that is very kind!

The scene in the movie in which Mike and Kay are having dinner takes place in a Manhattan hotel room.

The scene in which Michael has his own apartment in Manhattan is in the book only.

Michael takes Kay there after their reunion which, in the book, takes place at the Corleone mall.

When the book starts, Michael's in school at Dartmouth. When the Don gets shot, the family can't locate him immediately because He and Kay are in a New York hotel room. They're in New York for the holidays but taking their time about going out to visit the family.

As plaw said, the scene you referred to with Michael and Kay having dinner was in a hotel room where Michael was shacked up with Kay just before Vito got shot.
Sonny and his family lived at the mall at the time. The mall had a cluster of individual houses. Sonny and his family lived in a house separate from Vito's. But, in a deleted scene that followed the one you referred to, you see Sonny walking a very short distance to tell his mother that Vito'd been shot.
The movie never shows us where Michael lived immediately after the war. The first we see of him is at Connie's wedding, and he's in uniform, which could mean that he was living on an Armed Forces billet at the time. Then, as now, it was considered permissible for servicemen to wear dress uniforms to formal occasions like weddings. Or, he could have been living anywhere and simply decided to wear his uniform either because he didn't own (or have access to) a suit or a tuxedo. More likely, it was to reinforce his independence from his father by parading his decision to volunteer for the Armed Forces instead of entering the family business. You can be pretty sure he wasn't living in the Mall at that time.
As moustachepete said, the book explicity says that he'd returned to Dartmouth College to finish his college degree. He was staying at the hotel, with Kay, prior to visiting the Mall for Xmas. Since the movie shows him having dinner with Kay at the hotel, I conclude that he was staying there.