Is every finite dimensional semisimple algebra over $k$ isomorphic to a direct sum of finitely many matrix algebras over $k$?

is it true that $\exists t \in \mathbb N$ and $n_1,...,n_t \in \mathbb N$ such that $R$ is isomorphic with $\oplus_{i=1}^t M(n_i,k)$ as a $k$- algebra ?

No, the best example being, IMO, the one given by Geoff in the comments: $\mathbb H$. It is not a matrix ring over $\mathbb R$ because a nontrivial matrix ring over a field always has nontrivial right ideals (but $\mathbb H$ does not.)

can we characterize such semisimple algebras over a given field $k$ which can be written as a direct sum of finitely many matrix algebras over $k$ ?

This may not be satisfying, but one criterion could be to check that $End(S_R)\cong k$ for every simple right module $S_R$. That's a necessary and sufficient condition that all matrix rings in the Wedderburn decomposition can be taken to be $k$.