1) Yoshi is great in teams (Double Yoshi is bad though). He can't be gimped, hits hard (fair 17%, nair 14% etc.), best ledge grab, can wobble with partner pummel + dtilt, can wall an opponent out with eggs and short hop bair.
All his specials are great: up-b allows to prevent any aerial approach from the opponent (egg does 12%, explosion 10%), neutral B puts an opponent out of position, break their defense and make a 2v1 for a short period of time, especially if you egg lay a floaty character, and down-b can be used for team combos (grounded down-b kills a lot earlier and aerial up-b, and kill non-fastfallers at fairly low%).
I don't think it's needed to Yoshi to make double 1v1.
Yoshi can combo with his partner (that partner needs to know Yoshi a lot for that), wall out an opponent, support, ground approach with wavedash tilts, aerial approach with DJC attacks.
I feel like he just needs space from his partner, because of his movement options, having an ally close will make him accidentally attack his teammate.
2) Before telling about any characters, it seems Yoshi's up-b do great with heavy-grounded characters, like Samus, Luigi or Sheik. these partners don't go above the ground, and you cover the air with your eggs.
I can't say any character is better in all situations, it's up to the style of the players, one character being better according to the situation or matchup.
I've got success with Falco as a partner because he controls the stage, and Yoshi can go in and do his high dmg combos (don't forget Yoshi hurts a LOT). Also, Yoshi being a character that is fine alone against both enemies at once (because of weight, double jump armor and parries), Falco/Yoshi can to a formation with Falco lasers as support to kill efficiently an opponent:
PEYE (P = partner, E = enemies, Y = Yoshi)
Sheik is theorically fine because Yoshi can be the main attacker, and Sheik supports well. They can also do a PEYE formation with her needles or YPEE with his eggs support. However, while Yoshi can defend himself in 2v1, having a "ledge Sheik" is not the best for Yoshi since he can't get in easily unlike Fox, Falcon or other fast characters with good SHFFL. I think there are better partner choices, I didn't test enough with Sheik to tell if that's still a good choice.
I've also got success with Samus, but it's most of the time a double 1v1, and up-b egg support is superb with Samus' good ground game, so YPEE is the best with Samus. Yoshi can also defend while Samus comes back if knocked far away.
Luigi would be similar to Samus, except more team combos and needing to be more careful with up-b eggs when Luigi does a dsmash to aerial.
Because Fox can do anything, no need to describe anything. Fox can do Fox things and be fine with Yoshi (so nothing special) Only thing to remind is that fthrow would be the best throw to combo with Fox's u-smash (or other character's kill moves).
Falcon can give all the space Yoshi wants and still be available because of his speed. Also, Falcon is the best to save Yoshi because of his up-b giving Yoshi a jump back. Throwing eggs or neutral-b can also save Falcon with his predictable up-b recovery.
Peach can cover the air, allowing Yoshi to stay on the ground and doing tilts. She recovers slowly, and Yoshi can defend himself (like in the Samus part).
3) I'd say in doubles, Pokémon Stadium is the worst, Kongo Jungle 64 is bad, FD and Dreamland are fine, Battlefield and Yoshi Story are the best.
Dreamland isn't that bad because the platforms are high enough and placed so up-b eggs are good and wavelands are still useful, more space means less friendly fire attacks, and high ceiling allows Yoshi to survive a lot more because of the chaotic nature of 2v2. I got to ~220% two times in a game in Dreamland, allowing to win the set against better players.
PKMN Stadium is the worst because of how large the stage is, low ceiling and transformations being more difficult to be in with 4 players in the match. I'd maybe even switch myself to Falcon in that stage sometimes against two non-spacies if they go to that stage.
FD isn't that bad because it's less large than Stadium and no changes to the stage. Also, without platform, up-b pressure is better.
