Is this a counter-example to Darwin's theory of evolution?
Darwin cited the giraffe as an outstanding example of natural selection. Supposedly, as a result of extended droughts, the supply of green leaves could be obtained only at the top of the trees, and therefore the shorter necked giraffe died off.


The term outstanding is funny in its usage here but otherwise this text is quite right so far.


  However, there is no evidence whatever in the fossil record or elsewhere that giraffes with short necks have ever existed.


Actually there is. See this popular science article suggested by @James.


  And what would have happened to young giraffes with relatively short necks?


The answer in the text above. It rightly says the shorter necked giraffe died off.


  Darwin failed to realize that body characteristics in offsprings are determined and programmed by DNA factors of the Genes or the Genetic material of the parents [..]


This is an awfully phrased sentence making it very clear that the author has no idea about genetics.


  not by the stretching of the neck or any other bodily exercise


Sure. The concept that individuals stretch their necks, necks become longer and this characteristic is passed on to the offspring is wrong indeed. This is known as Lamarckism. It is funny because the above text describe natural selection and it then says that it is wrong by making a straw man, misrepresenting natural selection as a Lamarckist process.




  What is the current status of "theory of evolution of species" ?


You should have a look at the post Is evolution a fact?.

You should also have a look at an intro course to evolution. There are several online and free introductory ressource of information on evolutionary biology. Understanding Evolutionby UC Berkeley is probably one of the best. It is very short, very introductory and will already teach you a lot.