Do rights exist?
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First we must define what rights are. I will start with google's definition:
Right: A moral or legal entitlement to have or do something. 
I think it is fair to say that legal entitlements to have or do something exist. They vary depending on the country and one's circumstances but they clearly exist. This is boring though. Moral rights are a more interesting topic.
What exactly is a moral entitlement to do something?
Interestingly, google's (and other dictionary's) definition of 'entitlement' is "the fact of having a right to something." The definition of 'rights' is circular! They aren't defined in terms of something else, so we're going to have to make up a definition of what it means to be (morally) entitled to something, before deciding if such 'rights' exist.
Here is what I think it means to be morally entitled to have or do something (since I am basically making up a definition, I can give no strong argument to support it):
I think it is necessary that (A): the act of doing (or acquiring, having, etc) this thing (let it be labelled 'X') is not considered morally wrong. I think it is also necessary that (B): to deny somebody 'X' (when they desire or need it) would be considered morally wrong. If (A) & (B) are true, I think one is morally entitled to 'X.' 'X' would be a moral right.
To make this a more convincing definition, try substituting the words "moral/morally" for "legal/legally." I think this would make a pretty good definition for legal rights, and as moral rights are analogous to legal rights , I think it is therefore reasonable to accept my definition of moral rights. If you like you could think of moral rights as legal rights, but based on moral laws, rather than the law of man.
(Of course I'm no lawyer, so I could have missed something in my definition of legal rights - if anyone has a suggestion it could be added?)
Unfortunately this definition opens up a whole new can of worms when we move on to the main question of whether moral rights exist. For (A) & (B) to be true, objective moral laws (at least, laws concerning 'X') must exist. This has been debated numerous times here before, so I won't bother here.
My answer to the question will be this: If objective moral laws exist, then moral rights exist. If objective moral laws do not exist, moral rights don't exist.
(I had to choose a side to come down on, so I just chose the most empty side)
So, if objective moral laws exist than moral rights exists too.
First, I'll just say that I don't believe in objective moral laws, though I am sort of on the fence right now.
Could we know what these moral laws are? I think that depends. If, say, objective moral laws are dictated to us by some all knowing deity (that we know can't lie to us) then I suppose we might be able know.
Otherwise I don't think we could know what they are (If we didn't know what they are, we also probably wouldn't know that they even exist). The only reason I say that we wouldn't know is that I can't think of a method we could use to determine them with certainty.
Assuming that we cannot know what the objective moral laws are then I think you are right that moral rights couldn't have much relevance to us.
This doesn't mean I think we should descend into anarchy or something though; legal rights seem to work pretty well at keeping society organised, (and we can at least usually agree on what they are) which helps the most people be happy, which most people just intuitively think is a good thing. That's enough for me, we don't need to know what moral rights are to have the happiest possible people. (I know we don't actually have the happiest possible people, but we try)
Note that legal rights are not always a reliable way to help the most people be happy - slavery was perfectly legal for example, and in fact it was only people's perception of what moral rights were what changed that (I don't know any american history, so that's just what I've inferred from the tidbits of information I've heard about it, feel free to tell me the slaves were freed for other reasons).
Legal rights are often based off of what we think are moral rights, but for the most part they work towards our shared goal of happiness. We just make do with our limited knowledge of what moral rights are (if they even exist).
I realize the second half of what I've written in this response is probably full of holes - I'm tired and I find this particular subject quite hard to express my thoughts on. I hope it offers some insight.