For a turing machine which computes $y$ with argument $x$, how much does the descriptional length of a machine producing $y$ without input increases
First of all, there are two notions of Kolmogorov complexity, that Li and Vitanyi (in their classic monograph) call algorithmic complexity and algorithmic prefix complexity. Each of them is defined with respect to a universal computer which has a certain property.

Algorithmic complexity $C(x)$ has the property that for every computer $P$, if we denote by $C_P(x)$ the complexity of $x$ with respect to $P$, then $|C(x) - C_P(x)| \leq \gamma_P$, where $\gamma_P$ depends only on $P$. This property is known as universality.

Algorithmic prefix complexity $K(x)$ uses a computer in which the valid programs form a prefix code (a prefix computer), and we only require the universality property for prefix computers.

The actual definitions are slightly more complicated, since we also need to define, at the same time, the notion $C(x|y)$ or $K(x|y)$.

If you want to use some encoding of Turing machines as your definition of Kolmogorov complexity (of either kind), you need to use an encoding which is universal. But the point of Kolmogorov complexity is that which computer you use doesn't matter, since the resulting notions of Kolmogorov complexity only differ by an additive constant.

Finally, the Kolmogorov complexity analog of your "substitution" rule is
$$
K(x) \leq K(x|y) + K(y) + O(1).
$$
In fact, more is true:
$$|K(x|y) + K(y) - K(x,y)| = O(1). $$
This holds for prefix complexity.