Question on gym etiquette

Hi everyone. Two questions on going to a gym (the kind of place with weights and machines). I saw that many places say that you need to bring a pair of 'indoor shoes'. How strict are they about this? Do they literally need to see that you have another pair of shoes with you? And if so, what exactly do people wear?…. do they actually have another pair of sneakers that is reserved solely for use inside a gym?

And for women and appropriate gym wear. I understand that Japanese women tend to dress a bit more modestly – than say American women – at the gym…usually wearing a short sleeve top (vs a tank or a shell or a midriff baring top) that goes down to the hips or just below, and that they sometimes wear shorts over their athletic pants.

I (American) typically wear leggings, and a shell top that goes down to the tops of my hips. I don't wear anything that reveals my midriff, or my butt, or that shows cleavage. Should I be OK dressed like that at the gym (and then maybe just covering up a bit more en route to/from the gym)?

Thanks!