Given a non systematic linear code matrix representation $G_{ns}$ (k rows and n columns) we say that the codeword $c$ is equal to
$c=vG_{ns}$
(where $v$ is the input sequence)
Is it ALWAYS possible to transform $G_{ns}$ into a systematic code $G_{sys}$ (that is $G_{sys}$ can be expressed ad $[I_k | P]$, $P$ is a generic k rows and r=n-k columns) ?
If yes what are the operations we have to perform?
If we have now $G_{sys}$ and we calculate its related quantities ($d_{min}$, distances profile, multiplicities ) what are the relationships between these quantities and the same quantities calculated on $G_{ns}$ ?
Yes, depending on what you mean by "transform". For any linear code $C$, there is a code $D$ that is permutation equivalent to $C$ and has a systematic generator matrix.
Suppose $C$ has generator matrix $G_{ns}$. Let $G'_{ns}$ be the reduced row echelon form of $G_{ns}$. Swap columns in $G'_{ns}$ to produce a new matrix $G_{sys} = [I_k | P]$. Then the code $D = \{x G_{sys} \mid x \in F^k\}$ is systematic and permutation equivalent to $C$.
As noted in the comment by Git Gud, length, dimension, size, and minimum distance are all preserved.