I don't have a lot of time to watch the whole set from start to finish, so I can't comment a lot about play decisions in general, but a small comment about counterpicks for peach. Peach has a lot of great counterpick stages, but the only one on the random is dreamland and that's not her best (fd is also an option in some matchups, this one included). I'd be much more inclined to take a fox to mute city, jungle japes, or brinstar than any of the levels you used. Japes has the highest ceiling in the game and you can camp the side platforms with turnips easily, brinstar is almost impossible to get shinespiked on, also has a high ceiling, gives peach a lot of good recovery options since she can either go to the side platforms or skip them to go to the middle, and mute city's advantages for peach are well documented (though the lack of grabbable ledge is big - it's incredibly easy to edgeguard fox there, and he doesn't deal with cars too well either).
If you're limiting yourself to neutrals, fd is WAY better than fountain too. One grab on there if fox is above 45% or so, and fox should die.
Peach's best weapons vs fox are chaingrabbing, edgeguarding, and to a lesser extend turnips to disrupt his ability to approach. His best weapons vs you are upward kills and shinespikes. Your counterpicks should exaggerate your tools (removing or making it difficult to sweetspot the grabbable ledges, no/limited platforms to allow him to escape grabs, campable platforms, etc) and hurt his (high ceilings, varied recovery options, pits of acid that prevent kills off the bottom, etc).