You live in a home. You live on a street. You live at an address. They all essentially describe where you live, but the "in, on,at" is used to specify which location you are conveying.
to live usually suggests long term or indefinite residence; to stay implies short term, residence.
I live in London I'm staying at the Piccadilly Hotel
"Life" is a noun (the name of a thing. "Live," when rhymed with "sieve" is a verb (what we do until we die). "Live," when rhymed with "five" is an adjective, describint a person, animal, or plant...