Her fireball doesn't really mess him up much at all. You can just shield and run forward at long distance. Waddles come out a lot faster than din's so when you get a little closer you can hit her with a waddle and shield before din's can touch you (I haven't tried this for a while but I'm pretty sure it works). This means that Zelda has to approach you and Zelda's not great at approaching. DDD's range >>> Zelda's, especially ftilt. The fact that you can actually camp her and give her hell trying to approach with your range and speed are your biggest advantages in this matchup, imo - they allow you to control the match to some extent. Your two big disadvantages in this matchup are that you can't CG her and that DDD is an awfully big target for her lightning kicks. I wouldn't say Zelda owns DDD in the air, but her lighting kicks and uair are very dangerous. Basically if her feet aren't on the ground be on the lookout for a lightning kick and don't get above her because her uair is godly. Her dair isn't that good so DDD's uair, utilt, and even the occasional usmash are very effective against Zelda. If you are careful of her aerials, you can actually edgeguard Zelda pretty effectively. Overall I'd say this one is in DDD's favor.
In summary:
DDD can camp Zelda at close enough range, forcing her to approach.
On the ground, DDD's range makes life hard for Zelda.
In the air, Zelda's lightning kicks are pretty easy to connect against DDD
Neither character wants to be above the other.
DDD has better KO power and much better survivability.
DDD has an advantage, but not an overwhelming one.
Oh, and for stage picks, I like FD because it seems to emphasize your camping and range advantages. Battlefield makes it easier for her to pull off both her Din's and lightning kicks, plus it puts lends itself to aerial combat whereas DDD actually wants to be on the ground in this matchup.