First off, sorry this happened, but you are lucky that she left you her information.
1) How can I make sure that the appraiser is not going to low-ball me?
You can only work with them. Take it to your preferred auto shop for an estimate. Push back if the insurer doesn't want to pay enough for full repairs. Your car needs to be put back into its pre-incident condition.
2) The DRL of the impact side is not working but the headlights, high-beams etc are working fine.
If this is a result of the accident, it should be included in the estimate, so it is returned to you fully functional.
3) Can I still ask for a replacement headlight?
If there is damage to the headlight, yes, though I don't see any in the picture. They'll only replace what has been damaged.
4) How can I make sure I get a rental car as good as my IS-350?
Guaranteed, 100%, you will NOT get a rental as good as your IS. When my last auto body repair was done, I got an Altima, and I paid extra out of my own pocket to upgrade to that.
5) Do you think my bumper needs replacement? I can still fix it?
Hard to say from the pics. The auto body shop will tell you if it can be fixed.
6) Is it possible for me to file a claim for diminished value, if yes, what's the process? Can anyone recommend a lawyer in NYC area? What are my chances?
I have no experience with this so I won't even try. Though I have a hard time thinking you'll be successful, for such a small accident. It may seem like a lot to you, but in the grand scheme of accidents, it's small.
7) Any recommendation of a body shop in NYC area who can help me out with this?
I cannot help you with this, unless you want to bring it to Seattle for repairs.