Trust me anyone reading this, if you are not feeling well, or something is distracting you, please, please, do feel free to reschedule your interview, even last minute.
Once you go through the process, and it's a no-hire, that is recorded in the system and you won't be allowed to try again for some extended period of time. At my company, it's 1 year.
I think I too wouldn't have "dared" to postpone, in some/many cases
Also: Do you really want to date someone who doesn't care that your cat may have a life threatening problem?
Finally, even if it's not that serious (cat emergency): Do you really want to work/date someone who is that susceptible to this bias? I know it's the norm to fall for first impressions, but for me, it's also a signal of problems I'll have with them in the future.
BTW, my cat had a serious health problem once. She needed lots of immediate care for a few months (at home, away from work). And the vets were telling us that even if she got through it for now, it will come back soon and we should really consider euthanasia - the condition would eventually kill her - not many cats would last a year with it, and chances are her she would be in pain for much of it.
And this all started a few days before I started my new job.
I went straight to my new manager (before my official start date), and explained to him that the cat was my priority - I could take a leave of absence or whatever was needed, but I needed to figure this out (either euthanasia, or time off for treatment options, etc) and could not be anywhere close to 100% at work.
Fortunately he was sympathetic and said I should not worry and do whatever was necessary.
The job turned out to be crappy, and I stayed there longer than I should have entirely because of this.