i) I'm in that 'what am I going do after I graduate' place. What you do looks really interesting to me. What would you suggest to someone like me who wants to know more?
First off, it's cheering to hear that my work sounds really interesting. Most days I think it is too, which I guess indicates that I am in more or less the right place. But I'm going to interpret your question (reading it as "how do I get an interesting job like that?") in both a specific and a general way.
Specifically mine is not an entry-level job, and my employers are not hiring entry-level people for my sort of role. I am now 42, and I've got here by way of postgraduate degrees in completely different subjects, political activism, field work in post-conflict transition countries and eight years in Brussels thinktanks. I'd always hoped I might end up doing this sort of work, but the route to get here was not obvious.
Generally, it is worth investing some time in reflecting on what you want for yourself. The best book on this subject, for me anyway, is Richard Nelson Bolles' What Color Is Your Parachute?. Go and buy it, as soon as your nearest bookshop opens. You won't regret the tenner or so it costs, or the weekend you will need to spend filling in the worksheets.
ii) Related to the above, I'm hoping to do a Masters no matter what happens. Would an International Relations Masters or similar be useful, or would a Masters in a related discipline, specifically the War Studies MA at Kings be a good way forward?
Since my own masters is in medieval astrology, I am not sure I can give a terribly concrete answer. What you will get from a taught postgraduate degree is i) information about whatever it is about, ii) analytical and research skills, and iii) a few months' respite from the job market. I'm not convinced that academe is ever especially up to date in international relations; you have to rate the comparative utility of going to lectures vs reading a couple of decent newspapers every day. IR theory is almost completely useless in terms of the day-to-day practicalities of diplomacy. I have never actually studied politics and have no regrets about that; nor has it hampered me from following a political career. (I do sometimes wish I had a better grounding in law.)
The analytical and research skills are, however, important and transferable, and doing a Masters allows you to hone those skills in an environment where your early mistakes are pretty cost-free. So I must say if I was shopping around I would be attracted by a course which had a decent component on the nuts and bolts of research and writing, whatever the particular area of interest.
And then there is getting a job, where it's very true that just having any postgraduate qualification ticks one of the necessary boxes for a lot of positions (including the internships in my own office). I suggest you google the phrase "has just graduated with a masters in X from the university of Y" and see what comes up! I notice that I meet far more people who have studied at LSE than Oxbridge here in Brussels; I don't know if either is on your list.
iii) how do I track down appropriate internships to get some insight into all of this?
This is the easiest of your questions to answer. For British-based internships, check http://www.w4mp.org; for Brussels positions, check http://www.eurobrussels.com and http://jobs.euractiv.com - I shall be advertising the next internship vacancy in my own office on all three (probably advertising at end of August for a start date in October). Also if you identify an organisation that interests you, there is no harm in the informational interview and direct approach.
A big job market is about to open up in the European Parliament, starting this evening. Likewise at Westminster a whole lot of opprtunities may open up next year (and also people will already be jumping ship in advance of the end of term). All of that has an effect on the overall market in political jobs.
My other piece of advice, not exactly internship related but in the same ballpark, is to check out the prospects of election observation missions - that link is a little out of date but the basic information remains correct. Doing ten days in a transition country is a good way of testing your attraction to working in the field!
I hope all that is helpful. Feel free to take this up with me directly by email as well.