No comments about the lying thing, but as for her plopping downa nd joining you:
Feel free to just not engage. Don't be friendly, keep your eyes on your food, grunt your answers to any questions, or answer with the fewest words possible (think: Calvin Coolidge--I love purple's suggestion of 3 words or less; feel free to take your time to answer so you have time to decide which words to leave out to get it down that far). And maybe say, "I'm sorry, I like some alone time to decompress at lunch."
And conveniently ignore the fact that when someone you -like- sits down, you are more friendly and animated. You are not obligated to explain that in any way.
(and if she says something, like, "I thought you wanted to decompress, and now you're talking to him," you look puzzled and say, "I'm not sure what you mean. But anyway, excuse me, I want to talk to my friend.")