You put everything I have thought about Frank and Claire into words so wonderfully. Claire, by far, is the most interesting character in House of Cards. She is cold. She is ruthless. She is unforgiving. She is immensely successful — yet most see her as a villain. Guess who else was described in these terms, but very endearingly and with respect? Steve Jobs.
I think that this is what makes Frank and Claire’s relationship even more interesting. Frank knows this about Claire through and through, even when others are diminishing her position. And the falling out happened because he forgot that she is so unflappable in her pursuits. She is his equal. Others might not see it that way, because of how society perceives women. But he knows it. That makes Claire his only real adversary.
In the end though (and this is a little off topic), I think they do very genuinely love each other. It’s just a love that needs to adhere to certain standards of power and progression for both parties involved, and I genuinely wonder if it’s really all that bad. In some ways, I find that even more romantic. (I also wonder if I am just mildly sociopathic too so, you know, take what you will.)