Natural rights are those rights inherent to a being.
But how would a right be inherent to a being?
We can say that a person has a right to live. But that person can easily be killed; so that right is not inherent to her as a being. It is something that is bestowed into her by others.
Those rights are inalienable if they may not be overridden by social contract.
This is, I think, a tautology: if something can be overridden by social contract, then such thing is alienable, ie, not inalienable. So you are saying that an inalienable right is a right... that cannot be alienated. Yes, it is, but no new information has been added by this.
Without some sort of deity, I don't understand how inalienable natural rights can be defined in any way that is useful (legally).
I tend to agree with you. Either there is some kind of deity, that bestows "natural" rights upon us (and so, they would more properly be called "divine" rights, instead of "natural" ones - unless of course the deity in question is "Goddess Nature"), or there is no deity at all, and whatever rights we have are social, not natural. The latter seems quite obvious to me, but the former isn't, at this level of analysis at least, self-contradictory.
Our natural right would seem to be "do what thou wilt", and this right would seem to always be inalienable--we can always do what we want regardless of what social contract we have entered into.
We can do whatever we want, but society will sanction some of our actions, making them illegal. The "social contract", to use your phrasing, does not forbid us from doing anything; it just forbids us from doing some things without facing certain consequences.
This isn't useful though, because we want our social contract to be binding.
Then there are no such things as "inalienable" natural rights: if the social contract is binding, it can, theoretically, bind us to any arbitrary rule, overriding whatever "natural" rights we may believe we have.
It is a fact that in practice, the "social contract" cannot stripe us from some "rights" lest the society as a whole falls into disorder. But then the foundation of these "inalienable" rights is not "nature", but the internal logic of human societies (if murder isn't forbidden, then cooperation between humans becomes more difficult, if not impossible, and consequently society cannot "progress" into greater wealth and harmony).
From the perspective of one who wishes the well-being of all, we might consider that "life, liberty, and the pursuit of happiness" are great rights that all people should have, but I don't understand how we can reasonably define these as inalienable natural rights
I think they were historically defined as such, for historical reasons, and that this was a good thing at the time they were defined; but it seems quite obvious that they aren't "natural", and that they only are unalienable as long as our social frame make it impossible to alienate them. In the past, a person could sell herself into slavery, and nature would do nothing to prevent it. It was only a change in "social contract" (which isn't actually a contract, but for the sake of the argument, let's use the term) that put an end to slavery.
I further think that the deistic origins of the concept necessarily involve a divinisation of "nature". The people involved in the French and American revolutions might have not realised it, but they tended to worship "Goddess Nature" and attribute a divine role to it, instead of to the Abrahamic God. So again, no, you cannot have "natural rights" without undue and unaware attribution of volition and other human-like attributes to some super-human entity.