As Gray said, it's considered rude to ask a woman's age in the United States too.
1. No, I don't try to hide my age, but I don't announce it to everyone, either. I do sometimes joke about "being old."
Sometimes I make jokes about my grey hair - I like my picture with my grey parrot because he's the only one in our house greyer than I am! Sometimes I make jokes about being forgetful. Wait a minute, what are we talking about...?
In most circumstances, it isn't necessary for people to know my EXACT age. Does the person really need to know whether I'm 39 or 44 or 50? Sometimes even I forget. Forty-three? Forty-four? They feel the same. It's not like the difference between 17 and 18.
2. It depends on the situation. Who is asking? Why are they asking? Do they ask to find commonality, to understand me better, or to have information to use against me?
A few times, I have encountered older American women who will ask younger women their age and then afterward, they "use" the information like this:
"Oh, of course you're able to play music. You're YOUNG, so it's easier for you."
"Oh, of course you're able to speak Spanish. You're YOUNG, so it's easier for you."
No, lady. I can play music, speak Spanish, etc. because I take lessons and practice. I have taken lessons and practiced for many years. My "youthful" age of 44 doesn't magically give me wonderful skills. Please don't dismiss my hard work.