Do human rights exist?
Yes, in exactly the same way that law exists - that is, as a social construct that we human beings have come together to (mostly) agree upon. Declarations of human rights are living documents, in that they are open to interpretation when conflicts arise and are subject to change as we refine the language used to express the ideals behind them. Codes of human rights (or lack thereof) help shape the world, or at least in the places and aspects where humans spend the most time.
If you mean 'exists' purely in a physical sense, then no - you'll never be able to pick up a Human Right and chew on it. Human rights are (often codified) ideas, and like other ideas exist in a more abstract manner.
What is the basis for believing that there is such a thing as human rights?
At its most basic, the social contract (explicit or implied) that we human beings have with one another implies there is some set of rules we should all follow. There are many different possible social rulesets, many at odds with each other. The concept of human rights forms the basis of one possible ruleset that takes fair treatment of all human beings as a core concept.
In other words, human rights exist because we humans created them - in much the same way that art, music and philosophy exist because humans created those, too.
I am all for human rights but it could be argued that the notion of human rights is imaginary and reducable[sic] to absurdity (that humans by virtue of being born are magically entitled to certain things) when trying to prove it scientificaly[sic] or logically.
Human rights are not absolutes. Rather, they're our best attempt at expressing and codifying the golden rule. They are as imaginary as other ideals, laws, guidelines and best-practices - so again, whether they exist or not/are imaginary or not will depend on whether your definitions include other abstract concepts.
Logically, it makes sense for humans to treat each other respectfully (so that we are treated respectfully in turn) and to protect vulnerable segments of our population (so that we too are protected during times of vulnerability). Scientifically, we can show that the ideals expressed by the golden rule translate to happier people and improved productivity and quality of life - and assuming those are values we collectively agree on, we can continue to study ways to improve the situation for as many as we can.
It also seems incompatible with evolution, where survival of the fittest would dictate,
'Survival of the fittest' is a catchy phrase that is as accurate using 'It's just a theory' to dismiss a field of science. Wikipedia currently has a nice summary: (Emphasis added)
The phrase "survival of the fittest" is not generally used by modern biologists as the term does not accurately describe the mechanism of natural selection as biologists conceive it. Natural selection is differential reproduction (not just survival) and the object of scientific study is usually differential reproduction resulting from traits that have a genetic basis under the circumstances in which the organism finds itself, which is called fitness, but in a technical sense which is quite different from the common meaning of the word
In biology, 'Fitness' is a complicated, multi-parameter, interconnected mish-mash that is constantly being re-evaluated in response to an ever-changing environment that includes flora, fauna, seasons, social/sexual dimensions, etc. etc. etc. Using everyday definitions of the words, 'Survival of the fittest' is to biology as 'Things fall down' is to physics.
...that you either have what it takes to secure the resources necessary to survive, or you perish if unable to secure said resources, evolution doesnt state anywhere that everyone is born with a right to a specific alotment of resources.
Social animals (including humans) learned long ago that cooperation is a winning long-term strategy. Can you imagine how shitty life would be if we didn't have our modern division of labour? If nobody had time to program a computer because we were all so busy foraging food for today and desperately hoping we'll be able to do the same tomorrow?
Evolution (descent with modification) is an 'is', not an 'ought'. It happens to populations, not individuals. Selfish behaviours may help some individuals short-term, yet can be disastrous for the population as a long-term strategy.
Our history shows the evolution of the concept and codification of human rights from earlier social constructs, such as the earliest expressions of the golden rule right back in antiquity. There is nothing unnatural about an evolving social contract among social animals such as humans.
Is it possible to reconcile these contradicting positions or is it more logical to conclude that human rights are a human invention?
Yes to both parts: Once you move beyond a catchphrase-based understanding of biology (and the emergent human sociology) the conflict dries up like so much straw left in the sun.
That codified human rights are a human invention should be obvious, and their being a human invention - built by humans, for humans - in no way diminishes their worth.