To be pedantic, steel frames do fatigue, so they do "wear out" in a sense. People often don't get this because steel frames can often last a lifetime or two without any failure, and they often die some other kind of death before fatigue causes cracking. The idea that steel frames don't fatigue because steel has a fatigue limit is a fantasy fueled by people who have the romantic notion that steel is forever. So is the idea that steel doesn't fatigue because it isn't bent past the elastic limit, due to a misunderstanding of how breaking paperclips applies to bicycles. However this doesn't really change the material properties, it's just the source of frame cracks that show up and not caused by crashes or whatnot. The stronger the steel (in fatigue strength), the stiffer the frame (thicker tubes, smaller frames), and the lighter the usage (smooth roads, not being a pro sprinter), the more lifetimes a typical steel frame will last.

It might be worth noting that turbo trainers put a lot of stress on the dropouts especially if you try to sprint on them. It's a very rare but not unheard of occurrence for frames to break there on trainers. For a typical steel frame, this is a non-issue, but there are a few dropout designs that have a reputation for cracking in normal road use, and I wouldn't put them on a trainer.

As far as whippiness goes, if a frame doesn't feel whippy to you, then it's not whippy for you. Whippiness is not an inherent quality of a frame, stiffness is. Whippiness is when a frame is of inadequate stiffness to the rider riding it. If someone calls a frame whippy, either they mean they found the frame whippy, or they think the average rider would find the frame whippy.