1. Second Lincoln Ristorante which is literally AT Lincoln Center, beautiful, and used to getting people in and out on time. Also good for the "design your own Negroni" cocktail menu.
Other options nearby but less high end are Bar Boulud and Boulud Sud. Andanda for tapas. Blue Ribbon Sushi Bar and Grill for sushi and other Japanese dishes (not entirely traditional, but tasty - I actually prefer the non sushi dishes). Salumeria Rosi for more casual Italian.
2. Weekday breakfast or weekend breakfast/brunch? Sarabeth's, Good Enough to Eat, Nice Matin, Fairway Cafe, Barney Greengrass. The latter two are closest to "diner" like in ambiance. I have heard good things about Jacob's Pickles. Some may have viciously long lines on weekends (especially Sarabeth's, GETE, Barney Greengrass). Note that Barney Greengrass is closed Mondays.