My question is if it's possible to make organic molecules from inorganic.
Yes, it is. Wöhler synthesis that you linked to tells you it is possible, so I am not sure why you ask.
I oversaw the fact that urea was an organic molecule. I assumed that organic meant that it was a from a living-form, but I guess it doesn't mean that. It leads my question to go further.
What is the difference between the chemistry of atoms and molecules in living things versus non-living things?
I started to read a section of Felix Kaufmann's Theory and Method in the Social Sciences, but I found it very complicated for my understanding.
“During the early decades of the nineteenth century it was asserted that chemists would never be able to produce organic matter artificially (synthetically). A special vital force was thought to be required, an agent the creation of which is beyond human capacity. This theory was refuted in 1828 when Wohler produced synthetic urea. Soon other organic substances, like acetic acid and alcohol, and recently, hemin, a principal component of hemoglobin, were produced synthetically.
Nevertheless, vitalist did not admit defeat even though they had to make suitable modifications of their thesis. They now argued that the fact of synthetic production of organic substances in the chemical laboratory was to be conceded, but that the manner in which these synthese occurred was entirely different from their production in nature. The classic example adduced was the transformation of carbon dioxide into sugar, of particular importance in plant life. From the fact that yeast was indispensable in the process it was concluded that it contained the vital force required to start the process. But this assumption also proved untenable. Zymase, isolated from dead yeast, is able to bring about fermentation. It is now established that catalytic effects are involved, and, although the role of catalytic agents in chemistry is in need of further explanation, every chemist knows that catalytic processes are by no means confined to the vital sphere.
Besides this first biochemical group of vitalists, there is a second group that supports its thesis by reference to the specific mode of functioning of living bodies or their constituents, cells. They stress nutrition, growth, reproduction, and response to stimuli. In opposing them the mechanists tried to provide mechanical analogues of these processes. To this end, great ingenuity was exercised in the construction of ‘artificial cells’, hardly distinguishable from natural cells with respect to observable moments. The analogies, to be sure, cannot be carried very far, because the vital processes copied in this way certainly do exist in different interconnections, but they are not without significance, since they caution the vitalist against hastily claiming certain kinds of phenomena as peculiar to life processes.
We cannot discuss in detail here the vitalists’ various other arguments, which are intended to call attention to the special features of life processes, for example, their irreversibility or the phenomenon of death. But way of summary we can state that as far as the authoritative vitalists among biologists and philosophers are concerned, this kind of argument has receded more and more into the background, as almost every assertion of this kind was experimentally or theoretically refuted by the mechanist opponents. By contrast - and here we return to our point of departure - more recent vitalism does not see the characteristics of life in certain individual phenomena, but rather in complexes of phenomena to which, accordingly only a teleological approach can do justice. As the most eminent representative of this neovitalist tendency (which for reasons to be indicated immediately below can also be called psycho-vitalism) we should mention Deiesch, whose well-known arguments for the autonomy of life form the core of vitalist doctrine today.”