Well, I'm a lady, even when I am being a bitch. (Both the angry and sexual connotations of the word.) And I am a mere 22 years old.
I wear a hat, usually straw, except when I'm at home. I get a lot of stares, because most women don't wear hats around here. I don't take it off indoors unless I'm sitting down, because otherwise I'll put it somewhere and forget to take it with me when I leave!
I have met very few guys who look good in a baseball cap. I have met MANY guys who think tey do look good in a baseball cap. They should get a female opinion on that one!
As for shoes, I usually take mine off in my own home, and don't elsewhere, unless asked to do so. And if I haven't vacuumed lately, I'll warn you not to take your shoes off when you visit!