What are champions doing when they're not fighting in the League of Legends?
 

Are they off living their own life? Are they on stand-by? Are they confined for future use?

They are kept in Summonerballs

I would imagine that they're like pokemon. You call them when you need them, otherwise they're stored conveniently in stasis of some sort for easy access. Otherwise, Morgana and Kayle would have ended each other long ago.

I would have thought this, but their lore makes it sound odd for something like this to happen.

I lol'd

Ryze is perpetually standing in the unemployment line.

From the lore and the events regarding Urf, I'm pretty sure the League of Legends isn't just a battleground. It includes many areas, such as the beach Urf died in. And the magic that prevents all champions from permanently dying affects them as long as they are there.

Some champions aren't there all the time, though. Take Ezreal, for instance. He seems to be summoned only for combat.

Tristana has a house in the country with 33 cats and a vegetable garden.

shaco's lore kind of implies he just runs around the league shanking people 24/7.

Would not have pegged her as a crazy cat lady.