Applicability and calculation of Schwarzschild metric in General relativity for Earth Gravity As I found in this Wikipedia Article about Schwarzschild metric $$g_{00}=(1-\frac{r_s}{r})$$ where g - metric tensor.
I understand what is $r_s$. It is a Schwarzschild radius. For the Earth, it will be 8.9 mm. But what is $r$ it is not clear?
For example, what will be $r$ on the Earth surface (6371 km from the Earth Center) or at 10000km from the Earth Center?
Also, is it applicable to use the Schwarzschild metric to calculate the Earth Gravitational field (of course I neglect the Earth rotation. I hope the influence of the Earth rotation is very small).
 A: 
Also, is it applicable to use the Schwarzschild metric to calculate the Earth Gravitational field (of course I neglect the Earth rotation. I hope the influence of the Earth rotation is very small

The Earth is not spherically symmetric so that's a problem straight away.  It is also rotating, so that's another issue.
That said the effects of GR are measurable and can be calculated.  In particular there has been success measuring gravitational time dilation and the effect of frame dragging.  Note that frame dragging is only relevant in the case of a rotating body and is not modeled by the Schwarzchild solution.  To model frame dragging you need e.g. the Kerr metric.  Frame dragging is a very small effect for Earth.

Also, is it applicable to use the Schwarzschild metric to calculate the Earth Gravitational field

This may seem like a pointless statement, but it's perfectly reasonable to try any model as long as you understand the deficiencies of the model, which you seem to.  These GR effects are all very small for Earth.  Newtonian gravity works extremely well for Earth's gravitational field.  It's worth noting that the famous general relativistic correction for Mercury's orbit is smaller than perturbations from the gravitational fields of other planets on Mercury's orbit.  GR effects are, simply put, very small in the solar system.
