No and many lawyers who practice by themselves do not carry it.
Doesn't look like it.
Here... http://www.law.cornell.edu/ethics/ny/code/NY_CODE.HTM is the NY Code of Professional Responsibility. It doesn't seem to mention insurance at all.
No, attorneys in New York are definitely not required to maintain malpractice insurance. Neither the standards of professional conduct, nor any statute or other rule requires it.