Is it legal for a title company to ask you to write a letter stating why you are moving to their state, as part of their loan process?

Title insurance companies can ask why you are moving to their state since they will be on the hook so to speak in terms of any defects in the title that they insure. Hence, in general they have the right to ask, and you have the right to refuse. If this title company won't cooperate, insist upon changing title companies.

Sure. And it probably is not the title company, but the mortgage company underwriter who wants that information. Why would it be illegal?

I know of no law against doing so. The title company probably wants to know about your employment situation to verify income.