NOT A TEACHER
I would say that it's correct to say live not exist. If the people you are describing were so poor they actually died of starvation, they would still exist – as corpses.
In your second set of sentences, number 3 is most natural. With number 4, you could certainly say He likes riding bicycles, but not He likes riding bicycles every Sunday – it doesn't quite make sense. Something like:
He likes riding bicycles. He cycles every Sunday.
would be preferable.