Strictly speaking, Young's modulus is not always greater than the shear modulus, but it does tend to work out that way. You can see the reason why if you look at the relation between the two quantities (and Poisson's ratio).
$$
G = \frac{E}{2(1+\nu)}
$$
Combined with the knowledge that $\nu$ can be anywhere in the range $(-1, \frac{1}{2})$, one can see that G can be greater than E for $\nu < -1/2$. That being said, materials with such a negative Poisson's ratio are extremely uncommon, and it is safe to assume that the shear modulus is less than half of Young's modulus.
For an explanation of why the Poisson's ratio must fall within the above range, I invite you to check out some of my previous answers.
High-level: Range of poissons ratio
Detailed explanation: Limits of Poisson's ratio in isotropic solid
If I use the above relation, can I get Poisson's ratio at that
temperature? In other words, does the above relation hold true at elevated temperature too?
Yes, this relation is true for isotropic materials and the relation is independent of temperature, so it is valid at any constant temperature. This is why the experiments find values of $E$ and $G$ at various constant temperatures.
But, it's a relevant question, as argued here since the error induced in experiments that assume Poisson's ratio is constant while the temperature varies is non-negligible in most cases. You can see that in figure 8, for these composites, varying $\nu$ has virtually no effect on the Stress but does greatly effect the width strain and thus effecting the complex modulus.
There have been numerous experiments for various materials. See figures 8 and 9 here, figures 9 and 10 here and the figures here.
Depending on the material in question, the Poisson ratio can either increase or decrease with increasing temperature, however either way it is monotonic.
For more interesting materials, some work has been done to find generalized equations for $E$ and $G$ and then you can use your relation to find the Poisson ratio, i.e. composite materials.
And here's a nice nature article about modern materials and Poisson's ratio.
If you come across a paywall, email me and I can send you a pdf.
Best Answer
Shear viscosity is relevant when there is a velocity gradient in the material. The shear modulus applies in the static case.
For viscoelastic materials, both factors may matter. In principle, if you suddenly apply a shear to a material, there will be initial resistance to the motion due to the viscosity: once the molecules settle, only the elasticity provides a restoring force.
What differentiates them, then, is this: shear modulus refers to static phenomena, shear viscosity refers to dynamic phenomena.