I'm a native NYer - skip Ellen's Stardust and Carmine's. Personally, I'd give Times Square a miss too, unless you want to see a show. Agree that Via Carota is very good.
If you are looking for a special birthday evening, I very much recommend Bemelmans Bar at the Carlyle Hotel for atmosphere and jazz and a (very expensive) cocktail. Ludwig Bemelmans, who wrote and illustrated the Madeleine in Paris children's books painted the walls in exchange for his family living free at the hotel (image below). They used to have a collared-shirt dress code for men, don't know if that's still the case.
I love the Morgan Library Museum
https://www.cntraveler.com/activities/new-york/the-morgan-library-and-museum
For old fashioned southern food at a reasonable price, Miss Mamie's Spoonbread Too is fun - It's nice to go up and have a wander around Morningside Heights, where Columbia University is, and have a meal there. Also worth a visit - The Cathedral of St John the Divine and The Hungarian Pastry Shop - a true old school cafe hangout for Columbia students and faculty. The Met Cloisters, a bit farther up, is a lovely museum.
I love Daily Provisions (a small chain) at Amsterdam & 78th for breakfast sandwiches.
It's worth making an excursion down to Johns of Bleeker Street for a pizza. I'm also a huge fan of the square pepperoni slice at Prince Street Pizza (no seating, so it's lunch on the go). Joe's (no seating, slices only) in Greenwich Village is also hugely popular.
For Chinese food, I haven't been to this place in years, and it was closed but recently reopened - for delicious and very cheap food, we used to swear by Great NY Noodletown at the corner of Bayard & Bowery. It used to be kind of an institution - total no-frills but a real hangout for some of the most famous chefs in NY. Their shrimp wonton soup was amazing! And around the corner, Chinatown Ice Cream factory for fun, unusual flavours.
Pancakes - the best ones are at Chez Ma Tante in Brooklyn. The neighbourhood (Greenpoint) is a cool one to walk around if you're wandering over to the really trendy Williamsburg, but probably not worth going just for pancakes.
Gray's Papaya at Broadway & 72nd is a NY institution for hot dogs, it doesn't get cheaper than that (although full confession, I haven't personally had one since I was about 18).
The Doughnut Plant on Grand Street (lower east side) has fantastic and unusual doughnuts - they close when sold out, so hours can vary.