Will cycling off-the-trainer with the trainer's steel skewer damage my bike?

I assume you mean the quick release skewer and the answer is no, well at least not in my experience. I've spent a few years riding on the training skewer from my Tacx trainer and had no issues outside of the visual detraction of a silver skewer on black rims/wheels.

I have ridden several thousand km on the road with my trainer's steel skewer, QR and nut in use on an aluminum frame with no problems at all.

For training the weight just doesn't matter...realistically this is an aesthetic choice. There was a time when all these parts would have been steel.

Note if you ride in the rain your steel bits may corrode.

No, using a "trainer" skewer on the road has no negative impact on your bike or safety. The skewer which comes with a trainer is mecahnically and functionally identical to the one that came on your bike, with the exception of the shape of the "heads" on the trainer skewer. It is actually identical to one of the original skewer designs. The reason that one comes with the trainer is because most bike manufacturers have changed the shape of the skewer "heads" that come on the bike to be more aesthetically pleasing. With the side effect that the skewer that comes on your bike is unlikely to fit securely into the clamp mount on your trainer.

I do have reservations in using trainer skewer for outdoor rides.

My Elite Trainer Skewer is so easy to unlock, no matter how hard I tighten it. Just a little bit of pressure with just one finger is enough unlock the lever.

This can be dangerous, if your lever is unlocked by some object along the road.