While Michael's answer contains an awesome thought experiment, I disagree with its "fortunate" conclusion so much I think it leaves margin to a better answer. Also I'm not glad with basically all the definitions I've read there, except one - stoicfury's (read below). Also Karl's input (about the brain) is very insightful.
Contrary to Michael, I think definitions are very, very important. I'd argue it's fundamental for one's well being and it can lead a society to glory or doom.
I once had too loose definitions of words and no religion. So I was often confused with what I should think, with what I should do. Hell, I still have this problem today, and I will probably have to live with it for ever. As I began to give proper definitions in my mind, I also enabled myself to move forward, set a direction to move in my life. It's not much different accepting a definition I created myself or accept one from religion, but I think having it defined is very important if you want to build anything in your life.
As for a society, giving words to concrete objects is easy. And it facilitates communication to do practical things. "Pass me the hammer, please". Similarly, giving words to abstract concepts can help we orient each other to better position each one of us in society and build it up. "Should I get married?". "How many kids?". Yes, "kid" is quite an abstract concept. And I think the facility we have to communicate about it today is in part responsible for how big our Earth society is, despite the babylon of different languages.
So, allow me to give you my own personal definitions in hope to bring some new perspective here. I'm pretty sure they go against some common definitions, such as ones we can see in wikipedia, but those are how I see the concepts they're associated with, based on my social intuitive sense. Of course you can choose to use different words to try and add different definitions, give different meanings. But I think the current meaning people give to those words are quite broken...
Universe = every single thing we can even imagine
Some may perceive an idea / information / data as being something outside our reality. Dreams, for instance, can easily break rules of physics. Or can they? This is no simple concept and I won't digress about it here... Please, just take this definition and see if you can cope with it: there is nothing we can even think of that would be outside the universe. Not just the common concept associated with the "observable universe". There is no "multiverse". Everything is included. What exists, what doesn't exist and what might exist. If there is anything beyond the big bang that might affect us, it is within the universe. This is an important common ground definition I'll use for the other 3 you asked.
Free will = ability to think
I think free will has nothing to do with actions. I would say a restrained prisoner still have free will, even though he won't be able to do anything with it. A coma patient have no free will. Sure, thinking in itself can be considered work (i.e. an action), in a physical sense, as there are universe-embedded-neurons acting in our brains. And, as such, even our thoughts are restricted by the universe rules we live in. But, as long as we are able to rationalize anything, I call that free will. Maybe "free will" is just another word for "intelligence", after all. I love a definition I once saw for intelligence as "capacity to predict the future".
Randomness = events which were unpredictable
stoicfury's definition is spot on to me, just there's no need to say "any outcome is equally probable" as it is implied in the "unpredictable". To me "random" means "unpredictable". The moment some future event becomes predictable it is no longer random. If we live indeed in a deterministic universe (I think we do, but that's beyond the point), nothing is random by definition. Even then we can still perceive things as random while we can't predict them.
Non-determinism = events with unpredictable outcomes
What? So it's the same as random? Yeah, basically. A deterministic universe is a connected universe. If it originated from a single point in space time, the initial variables there and then have determined every single thing that will ever happen within it. Nothing in it is truly random. So a non-deterministic event is an event that will bring at least 1 random result. And indeterminism is just a prettier name for this same thing.
Conclusion . . .
Free will has nothing to do with randomness or determinism. You say random means the choice is unpredictable, but how could it ever be unpredictable if there was just 1 option to begin with? You're confusing definitions there. Unpredictable implies more than 1 option, more than one possible choice. Randomness and determinism are different ways to look at the exact same thing.
To all that, Michael's 3 tiny people experiment still applies, and the conclusion is still the same. Today (and maybe for ever) we can't tell with certainty if our universe is deterministic or not. Even if we define the universe as a non-cyclical infinite space-time continuum it still could be either random or intrinsically predictable. And the foreseeable future indeed make it seem like it is practically irrelevant to have a precise definition for those concepts.
And who's to say it really is irrelevant? So far, even if the universe is not random, we can't predict the future. I can only say, from past events, many definitions some philosophers did in the past are quite valuable today. I don't know why we like to argue about future concepts so much, but the value in defining all this may arise much sooner than we can expect. There is the very small chance we may live much longer to even see it happening. I don't think getting to a good and solid common definition is unimportant at all. Just the personal enjoyment of digressing about all those subjects are of quite enough value to me. Specially if I find someone who share the same view!
If our so called capacity of choice (often associated with "free will") is indeed an illusion, I find it very enjoyable how this deterministic universe can still uphold such a magnificent existential living experience. And finding a person who can agree, who got the same idea coming from a completely different point of view is kind of magical. It's probably the main reason why I go through so much trouble to write questions and answers here.
( made into a blog post for local internet backup ; original question and answer are from stack exchange philosophy and are mostly unchanged here ; topic photo looks much better on the big screen than I expected! )