Electrical – Does a power outage / unplugging damage major Appliances

You may not have trouble with your appliances, but the intermittent connection could be generating an enormous amount of heat and spark an electrical fire. If the intermittent connection is on the utility pole then you probably don't need to feel responsible; if it's in your breaker box or in your wall then you should be extremely concerned.

As a rule of thumb, a loose connection where your lights are either completely on or completely off isn't a concern; it's the times when your lights are half on where that connection can be dissipating an enormous amount of energy. Worst case, the connection could be dissipating on the order of one quarter of the full usual load, so with a 3400W electric dryer you could have almost a kilowatt of power heating up that connection. Could get entertaining fast.

I'm assuming you are in the United States for this answer. A stove is normally on a dedicated circuit which means it alone is the only load wired to a particular circuit breaker. This means nothing else would have been affected if your stove's breaker tripped. The breaker for your stove will be larger than a normal 120V breaker in your panel so you would have had to reset this if the breaker indeed tripped. If your stove was still on, resetting the breaker may have been difficult; the switch may have been difficult or impossible to switch to the "ON" position.

I am assuming you unknowingly reset the breaker when you were switching things in your breaker panel. If you are certain you didn't, then you may have never actually tripped a breaker and have some other electrical problem.