Thomas,

Quote:
If you are willing to forget logic for now, I am completely in agreement. Taking just what the Tanach says, Genesis 18 calls the three visitors "men". In the middle of talking to these men, Abraham is suddenly found talking to the Lord.


This is incorrect, and is amalgamating the verses.

Abraham isn't "suddenly" talking to the L-rd.

Quote:
Genesis 18:1
1. Now the Lord appeared to him [Abraham] in the plains of Mamre and he [Abraham] was sitting at the entrance of the tent when the day was hot


The L-rd is with Abraham at this point.

Quote:
Genesis 18:2
2. And he lifted his eyes and saw, and behold, three men were standing beside him, and he saw and he ran toward them from the entrance of the tent, and he prostrated himself to the ground.


Then he sees, and runs from his tent to the three men. Hashem is there before, during, and after the malachim appear on the scene.
I asked for strength, and G-d gave difficulties to make me strong. I asked for wisdom, and G-d gave me problems to solve. I asked for prosperity, and G-d gave me brawn and brain to work. I asked for courage, and G-d gave me dangers to overcome. I asked for love, and G-d gave me troubled people to help. I asked for favors, and G-d gave me opportunities. I received nothing I wanted. I received everything I needed. My prayers were answered.