1. Yes, foreigners who regularly communicate with Koreans decide to have a Korean name, either a real Korean name or their birthname that is just written (and sometimes shortened) in Korean. I think the latter is more common. One of my American friends' name was 제이. He had a long name and was hard for Koreans to pronounce. But he had 'jay' or something in his name. So he decided to be called 제이. 2. No, Koreans will feel more comfortable if you use a Korean name. And they will think you are a considerate person. 3. It is ok but I also think it's a bit too long. But until you have a better way, it's ok to use 알렉산드라. It's not too long, either. As you know, people often shorten their friends names to call them. So, if you keep your name as 알렉산드라, Koreans will probably just call you '산드라' when they become your friends. 4. As I said, you are not crazy at all. It is common.

Generally when introducing yourself you would be using your name in hangul. This isn't weird, because that's how foreign names are written! For example, Johnny Depp is written as 조니 뎁. It's probably harder for Korean people to read a romanized name, so it's actually BETTER to use that than just Alexandra. As for a Korean name for yourself, I've seen a few people do it on here, but I don't know how common it is. I don't see why not though, as many Koreans pick an English name for themselves, so why can't it be the other way around?

I've always wondered about this too. Maybe you can translate your name into Korean? Like my name, Melody, in hangul is 멜로디. Most Koreans I talk to online seem to not have a problem with that. ^^ Do you have a Korean friend or penpal? You can always ask them to come up with a name for you! I asked my Korean penpal one day and she gave me the name 신다솜(신 is the last name and 다솜 is the first name). ^^ They'll give you a name based on your personality! But I think just turning your real name in hangul should be okay~