There are some deleted scenes that refer to a previous relationship between Nathan and Elizabeth. In one scene, Nathan visits her at the refugee camp and asks her to leave Bishop alone. They talk about why their relationship didn't work out.
An extended version of the cafe scene shows that Elizabeth recognizes Nathan when he walks over to their table and is upset.
There is another scene with Nathan and Tom in their car waiting to meet the doctor, when Tom asks Nathan "Did you sleep with her?" Nathan doesn't reply.
There's also a scene that shows Troy Folger pressured Muir in Beirut to get rid of Elizabeth, so toward the end, when Muir is asked if he made that deal on his own, he and Troy both know Troy was the one who wanted her gone.
I think the story works much better without this love triangle piece, and I'm glad those scenes were deleted.

Of course they didn't. Nathan is CIA. He found out everything about her BEFORE going to the restaurant. The point was to prove to Bishop that HE didn't know enough about her.
620,000 whites died to free black slaves and not a single thank you yet.

I doubt it. I did not see anything to indicate they did.

I don't believe that they knew each other before, when they met at the caf I'm certain that her reaction would've been different to Nathan joining their table