I have it on my list of things to do to attend some Tea Gallery tastings. They have some simpler less expensive ones on their site, but really want to splurge for the more personal sessions (they have one for pu-erh, which I really want to do).
I don't know The Mandarins Tea Room. Will Google right now!
We do have a couple of ok cafes that serve loose tea. Cafe Grumpy in Chelsea has a very good pu-erh and several other loose teas. I believe they're from Rishi (I asked). And there's Think Coffee in the West Village which does loose teas. Karloff in Brooklyn (first time I had a Russian Caravan) is also great. There's a few other places where you can get a good white or oolong, and make a great alternative to Starbucks.