Choosing research area
This is one of the harder questions you'll have to ponder, and one of the most useful. First off, realize that your choices are not (completely) irrevocable. Many researchers shift areas every 5-7 years or so - the changes are not dramatic, but over time you can make useful contributions in a number of different areas.
Having said that, what you are looking for is a broad topic that will presumably take you through the 5-6 years of a Ph.D. To look at your questions one by one:
Where can I see if "getting my PhD to write software to guide probes to Jupiter" is even a viable option?
Do you know of people working on this ? maybe at NASA ? Where did they do their Ph.D ? what kinds of topics do they publish in ? What I mean is that one way to figure out if "Ph.D in X" is viable is to see if there are people doing X, and then figure out what their trajectory was by looking at their publications, CV etc
Must I choose a area that is currently being researched at the university I attend, or am I free to come up with my own?
It depends. You're in an MS program, and you haven't mentioned whether you're planning to apply elsewhere for a Ph.D. An MS is a good time to explore your options with people at the university, burnish your research credentials, and build some background in areas of interest. Obviously it helps if your area of interest is covered, but even if it's not, figure out related background that you need, and maybe look for someone doing research in a related area. Also see which places/people you'd like to do a Ph.D at based on the area, and that will give you a sense of what kind of extra reading/prep to do.
Can/should I be reading recently released papers on Computer Science topics for some inspiration, and if so where do I go to find those?
But of course ! but you should try to focus your search a bit. There's a large body of work in each of the topics you listed, and google is your friend. Once you find even a few papers, you can figure out where they were published, and then look at other papers in those venues, and repeat.
I know that I'm going to get downvoted for this, given the preponderance of computer-science people here, but my advice is:
Don't do a Computer Science Masters.
Quickly work out which are you want to do programming in, and switch to a Masters in that.
So, in your case, a Masters in Climate Science. Or rocket science.
Domain knowledge is really crucial for building useful models. Academically pure programming will help you write code to please the computer-science purists. But if you don't have the domain knowledge, it will be useless.
I've worked on many many dozens of successful applied programming projects. None of the useful models were written by people with postgraduate qualifications in computer science (but several of the useless ones were).
Alternatively, finish your Masters in Computer Science, and then do a Masters in Climatology, or rocket engineering, or whatever.
Then you'll have the best of both worlds: domain knowledge and postgrad-level programming skills. The best universities are developing in-house Research Software Development capacity, which seek to unit the two, and you could be part of that.
I would like to put in my two cents here. It looks like the type of areas you are interested in are related to Artificial Intelligence. I am not suggesting you jump the AI bandwagon but I do recommend starting to read on this area alongside your normal studies. Find some of the latest AI journals and skim through (at least) the abstracts to see what people are doing nowadays. Global warming and unmanned vehicles are huge issues. You would need to narrow this down to a particular area and make your contribution there. You can't solve world-scale problems all in one MS/PhD.
Read up and make a small contribution to the existing knowledge. Maybe you'll one day make a huge difference but, as the saying goes, even the journal of a hundred miles start with a single step.