No, it is surely not possible to write every general tensor in terms of 7 parameters. Clearly, the space of possible stress-energy tensors is 10-dimensional in $d=4$ (10 parameters), so you can't make it 7-dimensional (7 parameters). The special choices of the tensor that you mentioned are isotropic in a frame – treating $x,y,z$ on equal footing (they are invariant under an $SO(3)$). But general stress-energy tensors are not isotropic.

You may always diagonalize a symmetric tensor in $d=4$, i.e. replace it by 4 eigenvalues which I may call $\rho, p_{xx}, p_{yy}, p_{zz}$. However, the data needed to specify in which coordinate systems the tensor gets diagonal are equivalent to an element of $SO(3,1)$ – the Lorentz transformation needed to switch from a given basis to the basis of eigenvectors of the matrix – which has the remaining 6 parameters (the dimension of the Lorentz group), so if you also want to remember the information about the directions – and the fact that you included the components of $v^\mu$ shows that you do want to count them – then you're back to 10 parameters.

This is of course generalized to $d$ dimensions. A symmetric tensor has $d(d+1)/2$ components which may be decomposed as $d$ eigenvalues and $d(d-1)/2$ elements of an antisymmetric matrix whose exponentiation gives the right rotation or Lorentz transformation for which the tensor diagonalizes.

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