It is just fine. The gerund form (ing) just expresses an ongoing condition. In this case, you could have written the sentence as: I "CONTINUE TO HAVE HOPE THAT I WILL" do something. This is shortened by replacing the words in quotes with : "AM HOPING TO" which is a much more concise, as well as more commonly used, phrase.

Yes, it is correct. For some reason, "hope" can be used in the continuous. In this case, i implies you were thinking actively about doing something and had great hope of actually doing it.