When did the atom Theory Become indisputable? We know that the Greeks were the first scientists (or better, philosophers) to introduce the idea that matter is made up of little “chunks”, and they named these chunks “atoms”. This idea that matter comes in discrete small pieces was termed wrong and right over centuries when John Dalton, in the 19th century, constructed a theoretical framework in which he could explain in an elegant way the work of scientists such as Joseph Priestley and Antoine Lavoisier on qualitative and quantitative properties of matter and its transformations.
The framework which John Dalton used to better understand the work of other scientists had in its core the “language” of atoms, that is, he used the concept of atoms to explain, for example, why reactions only occurs if we have quantities “in proportion”. Note that the work of John Dalton doesn't proof that atoms are real, it only shows that thinking of matter in terms of atoms is a suitable way to understand the results coming out of experiments.
Scientists only proved that atoms are real at the beginning of the 20th century, and that discovery gave birth to Quantum Mechanics.

Now comes my question:
When did scientists prove that molecules “are real”? In other words, what is the history of the discovery of (the shape of) molecules?

Scientists have been playing around with toy models of particles for a long time but only a short time ago IBM researchers managed to “take a picture” of an actual molecule, and its configuration matches exactly the models we've been using for a long time, for example this pentacene molecule:



When did we first learn that molecules have a “geometry”? That is, an exact arrangement in space?
Short Answer: Henry Moseley provided the first experimental evidence for the nuclear model of atoms through X-ray observations in 1913. 

Long Answer:

Proving something is a rather nebulous task when there is no judge or jury and consensus is arrived at gradually over a century rather than instantaneously. Many scientists would consider the atom "proved" decades apart. 

Avogadro was the first to distinguish atoms from molecules, which alleviated much confusion regarding atoms and allowed quantitative description/experimental evidence as relating to Advogadro's number (a mole). That was during the early decades of the 1800s (19th century)

Various scientists became convinced at various times over the next century. 

Take Max Plank for an example. Around 1898 Plank became persuaded that the molecular approach was the only way forward for physics. He had formerly ascribed to the "pea soup" model where there were not discreet atoms. It was the molecular approach that allowed Plank to discover energy level quantization that is the foundation for quantum mechanics. E=hv

Lewis dot structures were some of the earliest electronic approaches to molecular geometry (circa 1910), but long before that there was an understanding of basic molecular geometries as demonstrated in the correct postulation of Benzene's hexagonal ring structure in 1865. Carbon was known to form 4 bonds with other carbons in the late 1800s. Linus Pauling was the first to calculate bond angles based on quantum mechanics in 1931.