They won't die without food, they just won't lay eggs. They'll eat when they're hungry, so you can just leave food down, but at a certain population/eggs on ground size they'll stop laying. They need space to lay, which means managing their numbers or making sure the eggs fall far enough to not be counted in their area.

The time between feeding and hungry - I've no idea specifically. A couple dozen can remove a stack of seeds in just a day or two. But unless you're cheesing the farming you need to clear out eggs regularly anyway so you'll be checking anyway. If you do want to cheese it there's info out there.