That's not true with properly applied undercoating. Rust requires oxygen and moisture. If a coating effectively seals out both of those, rust will not occur. Further, the underside of the vehicles is painted already from the factory, which makes it even less likely to happen, especially if the car doesn't have any rust on it. This is a matter of science and is really not up for debate. No water...no oxygen...no rust.

The surface must be cleaned thoroughly prior to any coating being applied. It also must be dry. Surface preparation is vitally important. If any part of the coating gets chipped, just like the paint on your car, you must have it touched up.

The likelihood, in normal driving, of anything chipping through an undercoating and then chipping the paint is slim. If you do a lot of off roading or otherwise are scraping the underside of your vehicle then yes, it can happen more often. Most rocks and stones will just bounce off.

Since you already have rust in place, it would be wise to remove it, paint the surface (sprayed like factory) and then undercoat for appropriate protection. If you just undercoat after sanding off all of the rust, you don't have the factory protection underneath the undercoating and it makes you more prone to rust.

I would highly recommend your undercoating be applied professionally to ensure accountability, warranty, and quality of workmanship.