정말 아쉬워요.

It's a bit light for a breakup. '슬프다' (sad) or '마음이 아프다' (my heart hurts) are better. However, you could say '헤어진 게 아쉽다' if it was a very casual relationship.

'아깝다' is specifically about the 'waste' of a resource (money, time, talent). '아쉽다' is about the 'feeling' of regret.

Yes, but use the formal form '정말 아쉽습니다' or '정말 아쉽네요' to be safer.

Use the pattern '~게 되어서 아쉽다'. For example: '못 가게 되어서 아쉬워요' (It's a shame I can't go).

Only if you *wanted* to like it but it fell short. If it was just a bad movie, say '재미없었어요'.

Yes, it is a common native Korean word used across the peninsula.

It's an idiom meaning 'asking for a favor' or 'begging,' because you only do it when you are in a position of 'lack.'

Yes, '너무 아쉬워요' is very common and sounds slightly more emotional.

It means 'Since it's a shame to end now, let's have one more drink.' It's a classic way to extend a night out.

Young people use '까비' (Kka-bi), which is a shortened form of '아깝다', often used in gaming.