Answering your last question first....
At Royal Street Veranda in New Orleans Square there is a Vegetarian Gumbo in a BreadBowl.
At Pacific Wharf Cafe in DCA there is a Vegetarian Chili in a BreadBowl.
Both, IMO, are delicious.
At the Trattoria in DCA there's a Caprese Sandwich that was VERY good. Rather olive-heavy, since it comes with a pasta salad with olives and the spread is olives (and I ordered a side salad which also had olives!) but all of the olives were good and I couldn't finish the whole sandwich it was all so good and filling.
The Blue Bayou restaurant (which *could* be an answer to number 3, for Most Unique Dining at DLR, if not "best") has a portobello and couscous dish that many like. It has a vinegar plate dressing that I went overboard with, so all I remember is VINEGAR from that meal, but DH liked it (we split the dish for the the-$4 split plate fee, and that got us one mushroom each and we each got a wedge salad). Speaking of that Wedge Salad, it's a quarter of an iceberg head, and a vegetarian will want to order it withOUT the sausage.
We found many options as ovo lacto vegetarians at both Lilo and Stitch and the Chip and Dale b'fasts (Paradise Pier hotel and Grand Californian). Extra bonus at the GC b'fast was vegetarian sauages! Others have reported them at Lilo too, but we didn't see them.
Have you looked at the menus online? They are at allears.net and dlresortinfo (the latter is better IMO), and can give you an idea of ingredients with the description of the dishes.
There are also veggie burgers scattered around...we tried one at The Hungry Bear, but it was cold and dried out, so I can't recommend it (though I've seen people's reviews with the opposite feeling!).
1. I think Goofy's Kitchen has both lunch and dinner. I think. The others are open for other meals, but are NOT character meals. Both PP and GCH also have menus to order from for b'fast, if you don't want the buffet. GCH stops serving off the menu at 11:30am, so if you have a reservation (priority seating) for that time, you will ONLY get the buffet, FYI.
2. Haven't had a character meal IN the parks yet. We have enjoyed both of the hotel-based character b'fasts. GCH had the veggie sausage and a beautiful setting, while Lilo and Stitch was happy and upbeat, Daisy Duck was there, and they had rice in a rice cooker (made my Korean DH happy).
3. Blue Bayou is very unique, though not always a fabulous *dining* experience.

You're inside the beginning of POTC ride, and it's just sooooo cool. I enjoyed the casual atmosphere at the Trattoria, and my server was great. She forgot to put my $6 pint of beer onto my check (the week after I got home I bought a 6 pack of the same beer for less than $6! lol) and when I pointed that out, she waived it, telling me it was her mistake. She got a pretty tip for that kindness.
4. Only food we had in DTD was Rainforest, sorry!
5. Those two restaurants I started the post with were counter service, and it's hard for me to miss them. Many love the dole whips, but I've seen the ingredients and they aren't pretty. And I think it tastes like pineapple flavored melted plastic. So I don't like them.

But I DO like, as a snack, a latte and chocolate chip cookie. Can get them many places, but for some bizarre reason I like them especially from the only outdoor vending cart that takes "plastic", just outside the castle on the left.
Sort of a hybrid of table vs counter is Plaza Inn. I had the kid's meal of pasta with marinara sauce, and a side salad, and it was sooooo good! The ranch dressing was perfect, and I'm picky with my ranch dressing!
Oh, and a similar not-quite-buffet but not-really-table-service but also-not-walk-up restaurant is Rancho del Zocalo, which also has a vegetarian option or 3. We liked our fajita veggie soft tacos in September, but during my solo trip I had a bean and cheese burrito kid's meal, and it was delicious.