1) I assume it was just a case of mistaken identity that wasn't really important in the moment - though the fact the the transponder was reporting the ship's name and class yet we never actually heard what it is seemed strange. Maybe there will be a throwaway conversation when Pike gets back. One could posit it was assembled from various parts collected from any number of battles during the war. That being said, the interweb says there was another one at Starbase 1,
NCC-1276, so it's more likely it's just another random variation.
2) The accent doesn't bother me - it's just her doing her "little old Jewish lady" voice (which is what she is I suppose). But as much as I love Kane, she talks soo slowly. It worked when she was playing Lillian on Kimmy Schmidt who's high most of the time; it works a little less here. Still, I think I'll get used to it - I could see it being related to the fact that she's seldom in a hurry since she's so long-lived.