I think I have an explanation for why some examples work and others don't. The phrase "at the drop of a hat" means quickly, readily, and without hesitation. There is also the strong suggestion that it is about something that happens right after something else. Let's look at some of the examples.
Ron replied at the drop of a hat. (He replied quickly.)
Henry is a very quarrelsome person. He gets into fights at the drop of a hat.
(It doesn't take much to get Henry angry.)
She took the job at the drop of a hat. (Somebody offered her a job and she accepted it without hesitation.)
I fall in love at the drop of a hat. (That person is very impulsive. )
Patrick gave me a diamond at the drop of a hat before I was going to bed. (It is hard for me to conceive of giving somebody a diamond quickly. Also note that in all the other examples the action occurred before something else. (BTW, he gave you a diamond before you went to bed.))
I went home and took a shower at the drop of a hat. (I understand taking a quick shower, but I don't so easily understand taking a shower quickly. The person would have to be in such a hurry that he doesn't bother trying to be neat but throws his clothes on the floor. What happened to make him want to hurry so much?)
She'll take her clothes off at the drop of a hat. (At the slightest suggestion she'll take her clothes off without hesitation. (She's very immodest. ))