Just to be clear, both forms imply the subject "I", so you are comparing "I hope" and "I am hoping". "I hope" would indeed be a common, more neutral way to put it, but "I am hoping" works too, particularly at the end of the letter, where it describes your friend's state of mind at the time she closes the letter. It's common in any event to wrap up a letter with an indication of the writer's attitude towards the correspondence, and if she used this as a closing, it could be an alternative to saying something like "yours truly" or "with best regards to the family" or whatever.

I wouldn't look too deeply into this, there isn't any difference between these two expressions. "(I) hope to see you soon" and "(I am ) hoping to see you soon" mean the same thing and can be used interchangeably.

This is a question about "hope" as a stative verb, isn't it? Yes, you learn that stative verbs don't usually take a continuous form. This should be your general, default option. However, we do see stative verbs taking continuous form, and in these cases, there's a very specific reason. In the case of your friend, it's temporary and at this moment. She's eager to see you soon. That's what a continuous form adds. "Hope" would be general, as well as a little formal and impersonal.