$X_t$ being a stochastic process, one cannot use ordinary calculus to express the differential of a (sufficiently well-behaved) function $f$ of $t$ and $X_t$.
Instead one should turn to Itô's lemma, one of the key results of stochastic calculus, which stipulates (assuming $X_t$ is here a continuous, square integrable stochastic process)
$$ df(t,X_t) = \frac{\partial f}{\partial t}(t,X_t) dt + \frac{\partial f}{\partial x}(t,X_t) dX_t + \frac{\partial^2 f}{\partial x^2}(t,X_t) d \langle X,X\rangle_t $$
where the quantity
$$ \langle X,X \rangle_t $$
represents the quadratic variation of the process $X_t$ over $[0,t]$ defined as
$$ \langle X,X\rangle_t := \lim_{\Vert P \Vert \rightarrow 0} \sum_{i=1}^N (X_{t_{i}}-X_{t_{i-1}})^2 $$
with $P$ representing a generic partition $\{t_0 = 0 < \dots < t_N = t\}$ of the interval $[0,t]$ and $\lim_{\Vert P \Vert \rightarrow 0}$ suggests the limit (when it exists) is taken in probability as $\max(\{t_{i}-t_{i-1} \vert i=1,\dots,N\}) \rightarrow 0$.
For Itô processes, that is, stochastic processes of the form
$$X_t = X_{0} + \int_0^t \mu(t,X_t) dt + \int_0^t \sigma(t,X_t) dW_t$$
or equivalently in differential form (this is an abusive notation, essentially used for convenience)
$$dX_t = \mu(t,X_t) dt + \sigma(t,X_t) dW_t$$
where $X_t$, $\mu(...)$ and $\sigma(...)$ are adapted (generally to the natural filtration of $W_t$) and the integrands verify the usual integrability conditions, it can be demonstrated (cf. any good stochastic calculus book) that:
$$ \langle X,X \rangle_t = \int_0^t \sigma^2(t,X_t) dt $$
or in differential form (this is an abusive notation, essentially used for convenience)
$$ d\langle X,X \rangle_t = \sigma^2(t,X_t) dt $$
In your case, because $X_t$ is the unique solution of the SDE
$$ dX_t = z_1 dt + Y_t dW_t $$
applying the above result with $\mu(t,X_t) = z_1$ and $\sigma(t,X_t) = Y_t$
$$ d\langle X,X \rangle_t = Y_t^2 dt $$
assuming the usual conditions are met.
As far as the second issue is concerned, as @Gordon mentioned, using the commutativity + bilinearity of quadratic variations:
\begin{align}
d\langle X + X^{'} , X + X^{'} \rangle_t &= d \langle X, X \rangle_t + 2 d \langle X, X^{'} \rangle_t + d \langle X^{'} , X^{'} \rangle_t \\
&= Y_t^2 dt + 2 Y_t Y_t^{'} dt + (Y_t^{'} )^2 dt \\
&= (Y_t + Y_t^{'})^2 dt
\end{align}
