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Young's modulus is determined experimentally by applying tensile strain (pulling on the ends) to a number of samples of the material under investigation and plotting the strain versus the elongation and taking the slope of the central part of the plot.
Yes, indeed. Sometimes tensile modulus is different from flexural modulus, especially for composites. But tensile modulus and elastic modulus and Young's modulus are equivalent terms.
The elastic modulus, also called Young's modulus, is identical to the tensile modulus. It relates stress to strain when loaded in tension.
Hookes law says that stress, s, is proportional to strain,e, as s = E e where E is modulus. Since strain has no units (it is deflection per unit length) the units of E are the same as s. E is the slope of the stress strain diagram.
Young's modulus