We have certain rights that should be unalienable.
Yes, everyone has certain rights inherent in their being a person.
What is the nature of these rights and what makes you think we have them?
That is, we could easily say that everyone is born with certain characteristics inherent in their being; A brain perhaps. Or DNA. Maybe consciousness. We know these things empirically or tautologically. We know that everyone has DNA because everyone we have ever tested has DNA; this is empirical. We know that every person has consciousness or potential consciousness because that's how we define persons; this is tautological.
What makes us believe we have natural rights?
Rights are valid claims of moral authority - the moral authority to act or think as you please. This moral authority stems from existence, ie you're alive (right to life) and whomever or whatever created you did so without chains (right to liberty).
Why does this moral authority stem from existence?
What you've said here seems a lot like deriving an ought from an is. We exist therefore we
ought to exist. That isn't logically valid; we would reject the argument "
North Korea is a dictatorship therefore North Korea ought to be a dictatorship". There's no clear connection between what is the case and what ought to be the case. Thus, where does this moral authority come from?
In any case, my personal opinion is that natural laws (and the natural rights which derive from them) do exist, in the atheistic sense that they were coded into our brains by a process of natural selection. The most simple example is "don't kill" - without which we couldn't live with other humans, even in the early days of humanity.
I'm going to bring up a similar objection here I'm afraid. Surely the fact that we're predisposed to act in a certain way does not mean that we
ought to act in that way? It seems very accurate to say that we are predisposed to a certain degree of xenophobia; one does not infer that we ought to be xenophobic. You've gone on to justify natural laws through the idea that they make human society possible. This is valid, to an extent. But it isn't a defence of natural rights; it's a utilitarian statement.
I do reckon that some rights are to some extent self-evident, since they stem from empathy, which is the ultimate source of our morality. It could be said to be inherently wrong to do harm to someone, when you would wish nobody to do that to you. Our brains are wired that way from a very early age, so we don't need laws and legally enshrined rights to see it.
Pretty much the same point again; How does it follow from the way our brains are wired (it's worth pointing out that this is hardly intractable wiring) that our brains should be wired that way?
People are created equal, and in general one person cannot be set above another. It is morally wrong to put people down and take action to deny freedom. It is not wrong to though inaction to allow inequality to fester (others may disagree).
This comes largely from the fact that people don't have an agreed upon purpose, so each should be allowed to pursue their own values.
I think that this is the very thing in question. That is, if your statement is correct it follows that we
shouldn't do these things, which is largely a rephrasing of natural rights. Consequently, I'd appreciate it if you expanded your justification of this statement.
As to "naturalness", I think this boils down to the question of whether rights are discovered or invented. I think the right answer is that they're invented in a way that involves so many logical and factual constraints that we might as well say that they're discovered.

The logical constraints come from the very ideas of justification, entitlement, and so on, which involve the parity of all parties to the discussion, i.e. all people. You have to propose ideas that it would be rational for me to accept and vice versa. As Souron says, this means that we have to allow each other freedom to pursue our distinct purposes. The factual constraints come from human nature: our moral instincts & such.
This appears to me to be a completely 'social contract' view of rights. That is, rights are formed through the negotiation and interaction of different interested parties. It follows that such rights are artificial insofar as they are man-made and certainly not self-evident.
Instead of explaining why they should be inalienable, it's way easier to say "just cause, dude! God said so." It makes the document in question a way better read. Imagine if instead we had a 20 page long philisophical explanation of why certain rights should be inalienable. "It's just the way things are, yo" wrapped up in eloquent language is way more marketable.
That's probably true. The problem then is the fact that since rights are justified by certain philosophical explanations they must rest on deeper axiomatic fact. I.e. Something we value more highly than said rights. There seems to be significant dangers in having your ideology and politics not built on what one most values but rather built on something derived from said values.
That said, natural rights are just as faith based as religion. A civil-religion. Just without the fantasy stuff.
An interesting point. The same issue applies really. Is it really best (and would it always be best) to rest secularised society on statements of faith?