Both are correct, but they might have different meanings. If A says to B "Write to me when you want" , this may imply that B is only going to write once. For example, if B asks if she should write a particular letter tomorrow or next week, A could say she doesn't mind when B writes it. Either tomorrow or next week would be fine. "Write to me whenever you want" might imply that B is going to write many times. A could be saying that she is happy to receive lots of letters at any time. Any time that B feels lonely, she knows that she can write A a letter. This is what the word "whenever" could suggest. By the way, you'll see that I've said "write to me" in my examples. "Write me" is used by American English speakers - you can say this if you want, but it isn't correct anywhere apart from the USA.

You are right, the correct way to say it is, "Write me whenever you want." People will understand you when you say "Write me when you want", but the proper way to say it is using the adverb 'whenever'.

It is correct but maybe you should ad "to". "Write to me whenever you want" and "Write to me when you want to". This is more understandable but what you wrote is also correct.