You should be able to find fasolakia, gigantes, horta, quail/ortikia, at Kalyvia, Mezes, Megas, Soula’s (owned by the Soula who owned Pan), Christina’s and Pantheon. I order and make most of these dishes. 
The Horta are any greens (Hortaculture, it comes from the Greek 
) - so Greeks use whatever they have, and cook them. Sometimes it’s a mix. Often dandelion, often spinach,sometimes arugula, mustard greens, beet tops, Swiss chard or purslane, often wild weeds, in Greece there are a dozen others.
The avgolemono dishes are becoming a little more rare. The avgolemono (chicken egg lemon with rice or orzo) soup is available at most full service restaurants. I guess my most recent order of avgolemono soup (Greek penicillin) was at Soula’s. Meat and rice dolmades served hot, with avgolemono are on a few menus, and some restaurants have cabbage rolls with avgolemono occasionally.
For the dishes you mention, I would probably recommend Mezes as your best bet for a places that would do all of these dishes at least fairly well.
Pantheon occasionally has rabbit as a daily special.
I had initially recommended The Palace, which was a restaurant that was still making dishes that are difficult to find, but it has closed permanently.
