I guess my last posting was too long for anyone to read so I'll make this one simple.
Can a company fire an employee for no aparent reason without documenting anything? This is in NY.

Simple answer. Yes. There are some exceptions, but generally speaking, yes. There is no law requiring any documentation, although the employer would be well advised to keep such documentation.

The fact that the employer has no documentation would probably (not definitely) mean that the employee would be able to receive unemployment. It would not make the termination illegal.