I should've posted yesterday, but after I clicked "Post Reply" the site went down for maintenance. I'll retype what I posted:
1. It depends on the matchup. I'd go YS against Sheik and Marth but not against Falco. I'd choose DL for Fox but not for Peach. In general, avoid FD since Yoshi is very good at utilizing platforms. Battlefield is more often than not a good choice in any situation since it's arguably the most neutral stage in the game. Each player has his/her favorite stages, so if there's one you really like to play on then I'd go for it.
2. Sorry for not answering you before; I wanted you to focus primarily on playing Yoshi and learning Melee since you stated that you were new to the game.
In any case, Yoshi's D-throw and U-air are his most potent combo tools. DJC U-airs work great on fastfallers like Fox, Falco, and Falcon for racking up damage at low percents (starting from ~20%). You can often get at minimum 2-3 U-airs on any character in the game at low percents. F-air -> U-tilt/jab reset also leads into U-air combos. Against Fox and Falco U-throw -> D-smash is a good finisher that works starting from ~85%. Also don't forget eggs, since well placed ones can lead into all sorts of combos strictly from the positional advantage they give.
Some of my favorites include weak N-air -> D-smash, last-hit B-air -> Nair, Egg -> N-air/U-air. Experiment and see what works (make sure to have other options available in case they DI favorably). Low angled F-tilt can also lead into U-air and N-air, but it's better as a disruptor. Here's a good example of F-tilt being used in a combo:
http://www.youtube.com/watch?v=d8ABeWc5_aI&t=7m9s
3. Are you recovering from the stage? If so, you can use Yoshi Bomb (Down + B) to grab the ledge (make sure you're not holding down by the ledge otherwise you won't grab it). It's extremely hard if not impossible to punish if the opponent is even a mid-distance away from you.
If you're being juggled above the stage, you have several options:
1. Absorb the hit with your double jump armor (that is if you're at a low enough percent where the attack will either barely break your armor or not break it at all)
2. Waveland onto the stage/platform
3. (as a mixup) Egg Roll onto the stage; Egg Roll resets your vertical velocity fairly quickly and you have a surprising amount of horizontal control over it in the air.
4. Yoshi Bomb if they misspace the aerial.
Against most characters you can just N-air and get out, but at times you'll want to mix it up.
