The chains typically use the engine oil for lubrication. Engine oil is critical for most other engine components as well. Some cars have another mechanism for the oil pump (other cars have gears, others are included in the timing chain, while still others have a separate chain drive for it).

The reason timing belts snap sooner than chains is, as mentioned, they're made of rubber with reinforcement mostly in the form of steel (like the drive belts outside the engine). They get brittle as they are used, and remember the tensioner is there to keep the tension up to spec. So in essence, if you open your timing belt covers, you won't "feel" any stretch because that's "masked" by the tensioner. When the belts reach the point of maximum stretch, and coupled with the tensioner and stresses of engine use, they simply snap.

Timing belts are usually maintenance free. Rule of thumb: when buying a car and there's no mention of timing belt replacement, have it replaced alongside drive belts (as well as the tensioner, cam seals, etc) especially if you plan to keep the car for a long time. Same goes with timing chains especially if it's really high mileage.