burnu sürtülmek

It's not exactly rude, but it is critical. You wouldn't say it to someone's face unless you are very close or trying to be harsh. It's mostly used to talk about someone else.

Yes, parents often use it when a child is being stubborn and eventually fails. 'Bırak yapsın, burnu sürtülünce anlar.'

'Pişman olmak' just means to regret. 'Burnu sürtülmek' implies a whole process of being humbled by external events.

Yes, it almost always implies some kind of failure, rejection, or hardship that forces the change in attitude.

In formal settings, you would use 'hatasını anlamak' (to realize one's mistake) or 'mütevazılaşmak' (to become humble).

Yes, 'Burunları sürtüldü' (Their noses were rubbed).

Only informally between colleagues. You wouldn't put it in an official performance review.

In many cultures, including Turkish, the nose is seen as the center of the face and a symbol of one's pride and ego.

Usually, it's for something significant enough to change someone's attitude. Not for just forgetting your keys.

Yes, it has been used for centuries and is well-understood by all generations.

Yes, 'Burnu sürtülmüş bir adam' means 'A man who has been humbled by life.'

Yes, especially in 'breakup' songs where the singer hopes the ex-partner's 'burnu sürtülür' and they come back crawling.