Here's my opinions on stages when vs. Fox.
Yoshi's Story-Worst platformed stage. Every top tier character seems to like it, and so many yoshi's like it. I know I'm probably going to get egg'd over this. On yoshi's, Fox's up-smash kills at rediculously low percents. Fox's up-air kills at low percents. aMSa chooses it a lot, but he's a very smart aggressive yoshi player and yoshi can do really cool tricks on the platforms. Fox can wall yoshi's out with a bair if you're off stage really easily and his sakurai combo is really good there. Fox also can recover quite easily and has wall jumps. Randall imo helps fox more than yoshi especially when it comes to edge cancelled eggs seeing how Randall can pop out and screw you up. I'd give it for this matchup, a 5/10.
Fountain of Dreams-Best Platformed stage against fox. High ceiling, unpredicatable platforms (better for a yoshi who is comfortable in wavelanding), and great for gimps since many attacks knock fox off-stage really quickly. Fair goes through low platforms and a full-hop fair can hit a foe standing on the top platform. Your up-smash can also go through platforms. I feel yoshi is a character who needs waveland-movement as well as crouch cancel abuse. Short leveled platforms can stop fox's SH dair. I give this stage 10/10.
Pokemon Stadium-I'd never pick this stage. The ceiling is too low, stage transformations are good for eggs, but the benefit isn't that great unless you're a good egg camper. Up-smash goes through platforms on neutral barely. Egg roll on this stage can work really well on Air (windmill) transformation. Again, the CEILING IS TOO LOW. I give this 3/10.
Dreamland-Best wide-ground stage. You'll be Ko'd at high percents. If you're fast enough at wavelanding getting laser camped is not a problem (this might not be the case with Falco). Down throw->upair combos can end with them DI'ing onto a platform which can make the combo chain longer for yoshi. There's also this thing I'm pretty good at where I fast fall and fair when I'm knocked really high by an up-smash and DI to the right or left. Basically a fast-fall fair covers a lot of space-time. If you anticipate the fox's jump to try to bair you (which I assume 100% of the time), FAIR REALLY EARLY and begin to fast fall, you may just get an easy kill. I can't stress the EARLY part enough. This a quite reliable strategy. I see many yoshi's only fairing when they can guarantee the fox is going to be grounded. Let's say your fair misses, who cares? You probably scared them from jumping and you still have your airdodge or egg roll for the little hop you may need. The chance of them punishing a fast-fall fair is slim if done correctly.
I give this stage for the fox matchup a 8/10.
Final Destination- Don't pick this stage either FOR FOX, for other characters this stage is fine. No platforms for maneuvering. I don't believe eggs can go all the way across and if i'm wrong, it's still probably not safe constantly doing full-distanced ledge eggs. The ceiling is not too low, but usually at this stage i find foxes liking to up-smash a lot at. Great for down-throw up air chains but not as nice for an agressive yoshi who should use platforms to his advantage. I can't speak about laser camping but I assume it's really good for foxes on this stage. I give this 6.5/10.
Battlefield- Great stage for many reasons. Good platforms, good ledges that punish a sloppy fox's recovery, and a not-so-bad ceiling height. There is no randall and my fair trick can work here quite well. I do go for quick gimps with forward and back throw on this stage and if you and fox are below a platform they probably won't jump so you can get the grab and when you throw them Randall won't catch them.
I give this stage an 8/10 like dreamland.
:D
(((((I am a defensive yoshi who enjoys parrying, edge cancelled eggs, and going deep for gimps but never being stupid enough to suicide. In my opinion double jump cancels are an unviable way to play yoshi unless they are coming after a grab or are being used in a tech chase scenario, so if you're the agro-djc nair on shield yoshi like many I've seen, this won't apply to you. I like leffen's yoshi a lot and some of aMSa's style is not my type. I hate trading/getting killed at low percents/being hit out of super armor/ taking chances. I crouch a lot and take advantage of Yoshi's bulk and usually live to 100+ on most lives. I'm not reading what OP said as this opinion is my own, feel free to critique.)))))