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I was thinking exactly that throughout this story. 
Without a doubt Iāve broken toes on at least two occasions that I recall, probably more. In fact, two nights ago I bashed my toe and it was bleeding all over the (white tiled) kitchen floor.
Iād never even think about going to the emergency room.
Stubbing or spraining a toe, or injuring oneself through clumsy accident is not the same as what the OP describes. In this incident the host describes a guest operating a light fixture properly and for itās intended use. It falling from the ceiling is almost certainly the hostās responsibility. Unless itās a defective light fixture, it was a maintenance issue and the hostās responsibility. Iām sure if she was asking their liability insurer to cover costs they would expect proper documentation from a Dr. or clinic.
Iāve stabbed my foot with a lawn aerator, been bitten by dogs, and tripped on a curb in the middle of street and not gone to a DR or clinic. But I quite possibly would have if I were the guest in this case. When a neighborās cat entered my home through a open garage door and bit and scratched me in the wee hours of the morning, I went to the emergency clinic. I didnāt even ask the woman to cover my co-pay but I wanted to make sure Iād done everything right to have my insurance cover in case of a serious problem.
What do you mean you doesnāt want the cat! Itās her pet. I bet she tripped over the cat and got injured and now sheās blaming you and leaving the cat behind. Make sure you mention to the insurance company that she had a cat.
Youāre joking, right?
KKC, No, she could have tripped over the cat and then blamed the light fixture. She could have easily tampered with the light fixture to blame the host. I find it fishy that she went to the ER for a bruised toe. Youāre be surprised how many con artists are Airbnb guests. Especially if itās here in the USA where people love to sue instead of working hard for their money.
Iām one of those horrible people who thinks the best of everyone until proved otherwise so I donāt think that there are many Airbnb con artist guests. (Iāve never come across one, anyway).
I think that when @muddy and I were saying that a bit of a bruised toe is no big deal we were trying to make @Perimaud feel better and that it wasnāt a serious injury. 
Okay. I acknowledge thatās a possibility but it seems extremely far fetched based on whatās been posted here. This is a shared room in someoneās home. And despite our reputation as a nation full of con artists (lead by the Don the Con) and litigators, I believe 98% of Airbnb guests are basically good, honest folks. I also think a scammer could find a much easier way to rip someone off that the narrative you suggested with the cat. But crazy things do happen.
As a teen and young adult I worked in NYC as a legal secretary. I guess that made me suspicious of most and jaded : (
LOL. I bet that would do it.
@KKC I agree that it was the hostās responsibility and that it would have been a shocking thing for the guest to experience. I just donāt think everything has to be addressed by offering money, but I realize that seems to be the American way. An "Oh my god, I feel so terrible, are you alright? ", running for an ice pack, and perhaps a gift of some kind- a bottle of wine, a box of chocolates, etc, would be more in order, IMO.
Everything? Did I say everything? No, not everything.
This thread (as is typical of the forum) ranged across the spectrum from what other hosts would do if they stubbed their toe at home to bizarre cat conspiracies. I was simply trying to return focus to a hostās responsibility when some defective item in their rental injures a guest.
I wouldnāt offer the guest money right off the top until I felt I had a good understanding of what was going to be involved. But also think host attitude can play a role in how the guest feels about the experience. And I will definitely be covering any costs that arenāt covered by the guestās insurance, my insurance or Airbnbās coverage. The guest should be out zero dollars for this injury.
In addition to the motion sensor battery powered lights I have a wall switch on each side of the large closet. Recommend a lot of light for safety.