is young's modulus of a material different for various diameters of that material? I would assume it does, but I cannot find a reference. Is the young's modulus of 1mm thick steel the same with 10mm thick of the same steel?
The young's modulus is like the 'spring constant' for a material. It comes from treating the atoms in a material as harmonic oscillators. It is a material property that does not depend on geometry. The young's modulus for both of the materials you mention is equal.
Now the stress they feel will be different under the same applied load.
No, Young's modulus is an intrinsic property of the material (e.g. steel) that does not depend on its form (e.g. wire diameter).