Because of unforeseen circumstances.

That's true, but he also does seem to genuinely care for Tyrion. This was displayed in both his stepping forward to fight for him at the Vail (despite saying it was just because he didn't have anything else to do) and his convincing Jamie to go see him in his cell when Jamie wasn't going to.

Because Tywin said repeatedly that the next whore he caught Tyrion with, he'd have hanged. Surely he'd notice Tyrion leaving with someone - someone who he'd easily discover was a whore.

As Varys said, when has Tywin Lannister been known to make idle threats?

That was never a possible scenario for Tyrion. It was a scenario that Shae dreamed up, but a man of stature (pun not intended) and responsibility cannot just pick up and leave. Tyrion may have been the lowest of the Lannisters, but as Tywin pointed out on numerous occasions, he's still a Lannister.

I'm pretty sure there's something going on between Shae and Tywin that we haven't seen yet.