CS is better for MK than FD hands down. FD is probably his worst stage overall: it's not large enough for him to play keep away/lacks platforms to save him from anything strong like Falco IC's, no extreme boundaries or features that he can abuse, etc. He can abuse how large the second transformation is, or the platforms on the first part. Not to the same extent as BF or similar stage but it's still an improvement over FD. It is one of his weaker starters though, especially if you encompass stages like Delfino Halberd Lylat.
CS is also a better legal stage than Delfino or Frigate IMO.
I disagree. The third part of CS is *essentially* the same as FD so I'm not going to talk about that part too much.
The second part is, essentially, negligible in most MU's since most characters can play keep-away against almost all other characters there (unless you're like DDD or Zelda or something). So you say he can "abuse" how large the second part of the stage is, but I really just see it the same way I view the fire and rock transformations on PS1; it's just a break in the match. Now if MK does approach there, he runs the severe risk of getting GR gay'd. Marth has GR->everything, Sheik/ZSS(inb4lolZSSgrab)/probably others have GR to regrab walk-off stuff. I'm sure a lot of other characters also have GR->stuff there as well. MK has... an earlier killing SL? That's about it.
This is left with the first part. The platforms are kinda blah for MK imo, especially compared to the benefits that other characters can get from them. The boundaries are fairly small which allows for earlier kills on MK as well. The stage is also very small which really hinders his ability to zone effectively and takes away a lot of chances for him to land safely with nado or out of a glide.
While there may be characters that do better vs MK on FD (Fox is the biggest one I can think of), overall I feel as though CS is still a worse stage for MK compared to FD.
CS is definitely a better stage than Delphino/Frigate/Halberd/YI:B though.