Stress vector - Stress tensor Is the definition of the stress vector the following?
The stress vector is the force per unit surface. 
The stress tensor is the matrix $\{\sigma_{ij}(x,t)\}$ and its $(i,j)$-component is the $i$-component of the force per unit surface that is exerted at an element of the surface perpendiculart to the direction $j$. Is this definition of the stress tensor correct? 
Which is the form of the stress tensor at a calm fluid? 
Is the definition of the (static) pressure the following? 
The (static) pressure is the diagonal entries of the tensor matrix. 
 A: Your definition of the stress tensor seems correct.   If you slice through a stressed body, the stress tensor has a component vector acting across the cut surface.   That's sometimes called the 'stress vector' but is better called the 'traction vector', the word 'stress' being reserved for the tensor.
Aside: in Voigt notation, stress is represented by a column matrix also sometimes called the 'stress vector'.   This is purely a notational convenience to allow us to write the (4th order) elasticity tensor on a flat piece of paper;  stress is properly a (2nd order) tensor. 
A: The definition you have is correct. A tensor has two directions, one on which force is applied and one on which measured / felt. When {i, j} denote the same direction we have a normal stress. When different, a shear stress.
Vectors are commonly written in a column, two normal and one shear, and  tensors in a square matrix.
The situation is essentially same in fluid mechanics and mechanics of materials. In the latter case when same hydro-static stress acts along three directions( written on main diagonal elements keeping all remaining shear elements zero), no failure can occur. It is essentially shear action in Shear or Von Mises stress  modeled on these theories for  failure. 
EDIT 1:
In  theory of Plasticity this failure causing difference component is the Deviatric stress tensor.
A: The above comment is correct for an ideal fluid, meaning the fluid has no viscous effects which is very idealized. For other constitutive models of the Cauchy stress for fluids the simplest form involves a relation that is linear with the stretching tensor which is how the Navier-Stokes equations are derived. 
There are also more complex non-linear constitutive process such as Rivlin-Erikson fluids.
