My Mom had to go to a nursing home after a fall. Do I have to give her assisted living home a 30 day notice?

And another thing. When sh dies, Social Security will demand to be reimburse her last month too. This happen to friends dad who die on Jan 25--not even 2 wks later arrives a letter informing his daughter.

Yeah, you may be stuck paying another month. Check the rental agreement.
This is something every one should know about, before signing any type of rental contract.
The facility can only show it to prospective new renter after it is empty out, carpet cleanes, repainted, etc. Even if they have a wait list a mile long, anybody new isn't going to move in without at least a couple weeks notice.

Where my mom lives, the new owners (Five Star Senior Living) put in a 60 day notice period, which I think is ludicrous. So when mom goes, I'll still owe $7,200 for two months of rent that she won't use. So 30 days seems good to me!

That being said, with NH, it seems to be different, based on the census of the home. Every time mom goes into the hospital, the business manager of the NH calls me and asks, "do you want us to discharge her?". the first time I was very suspicious, but in talking and listening between the lines, they were trying to save us money. So each time mom gets hospitalized, we dishcarge her and readmit her when she's ready to return.

Pretty standard, in my experience, the 30 day notice thing.

Also, if her belongings and furniture were still there in September, they weren't able to ready it for the next tenant until it was empty.

Read the contract. Most places do require 30 days notice.