If that is all the damage that you have done to your first strop, then you are doing better than when I started. As RezDog said, you can sand out small cuts, glue down flaps, or trim them out and sand them.
Check the toe and heel of your blade and make sure you didn't roll the blade where you hit the strop edge. A rolled edge can cause a lot of pain.

For oiling, any razor that I'm not using within the next 3 days gets oiled and the razors I use most often get their pivot oiled once a week.

I do store my razors in the bathroom, but I don't steam up the bathroom.