Question

1. I want to ask you a favor;would you lend me your note?
(Are the words 'favor' and 'note' properly used in this sentence? )

2. This is a private hospital or residential home especially one for older people. What is this? It's nursing home.
(Are the sentences correct? Does one mean 'a private hospital or residential home'?)

Answer 1

#1 is correct.

I would change #2 to read "this is both a private hospital and residential home for older people"
This will clarify your meaning.