That's it.
A cast iron skillet is one of the toughest objects in your kitchen as is, you don't have to baby it, just make sure it doesn't rust, and the seasoning will build up naturally as you cook.
I put mine upside down in the oven to dry. We always did that when i was a kid because the pilot light would provide a tiny bit of heat to dry. Everything is auto ignite now, but i still do it and think of my mom and grandma every time.
Can you explain why leave the skillet upside down? I thought the temperature of the oven should be pretty evenly distributed, right?
Hot water and a sponge. Use a tiny bit of dish soap if there's stubborn residue. If your cast iron pan lost its sheen, it's time to season it again.