Is the word order in the sentence from the letter correct?
"Write me something about you and your hobbies and what is your news."
The last part is a question, but it sounds strange to me like this.
""Write me something about you and your hobbies and what your news is."
It's not necessarily "write to me." It could be a teacher giving an assignment. "Write (something for) me..."
"Write to me" is about sending someone a letter or postcard.