Why is it appropriate to ask G-d to change nature i.e. healing those that the Dr’s have given up on etc.? And why doesn’t G-d usually fulfill such prayers?

Our rabbis teach us never to despair or give up hope from God’s mercy! It’s not only an exercise of belief in God’s ability, but also good advice, in the words of Vince Lombardi, “Quitters never win, winners never quit!” David HaMelech writes in Tehilim: “I searched … from where will my aid come? My aid is from Hashem who made the heavens and earth” (ch. 121). He is really the One and only who created nature, and the only One who can also change it. Even though usually nature functions according to the laws of His original “program” (God's minimizing His supernatural intervention enables us free will), sometimes He does intervene and makes miracles (most of us have heard of hopeless cases, surprisingly being cured, and not from medicine!). In addition, many questions regarding God can be understood with the help of the Father-son parable. The Sefer HaIkarim explains that there are 3 categories regarding what a parent will grant a child. Some things the parent will grant even if the child doesn’t ask. Other things the parent will not give, even if the child requests. The third category is that the parent may give the child his wish, if the child asks nicely. Being that we don’t know what issues are in which category, we always daven and pray, for often it may be in the 3rd category. In addition, speaking to Hashem brings us much closer to Him, gives us security, helps us define our goals, strengthens our respect which brings us to want to emulate Him, etc. etc. In short, prayer is not just to grant our requests. If your’e asking about a particular case, may I join as well to wish him/her a refuah shleima b’toch sha’ar cholei Yisrael! If that person is suffering terribly, then others, aside from his children, can pray that God put an end to his misery.