**Forward:** This answer does not address the main question "which statements are preserved for every generic extension." Which I suspect has no simple answer.

**Lets start** by defining what this answer is meant to address: Preservation results fall into two types (modulo fine details):

Large structural preservation.
An example of this kind is Shoenfield's classic result concerning the absoluteness of $\Sigma^1_2$ statements. These generally take the form: For some statement(s) $\varphi$ and "canonical" inner model $M$, we can show that $ZFC \vdash $"$\varphi $ is absolute for $M,V$ ". This is rapidly transformed into a forcing absoluteness result by cleverly setting up the notion of "canonical" so as to guarantee: if $V \subset N$ are models of $ZFC$ and $V\vDash M$ is "canonical", then $N \vDash M$ is "canonical." (In the case of Shoenfield's result this last bit is trivial since $V\subset N \implies L^V = L^N$.)

Small combinatorial preservation.
These results are numerous and deal with individual statements assumed to hold in the ground model and then using this assumption produce that the statement is true in the extension (using either the definability of forcing or constructing explicit names of objects in the extension, or a combination of both.) Moreover, results of this form can be considered as the reason a certain statement holds in an extension. These types of proofs can be understood as being essentially internal to $V$.

The contents of this answer will attempt to illustrate two examples of the second method and explain why $AC$ is preserved when passing to any generic extension.

**Now for a horrible and vague answer to a strange question:** What does using a partial order $\mathbb{P}$ to force over a model $V$ actually do? It lets people with imaginations strictly confined to $V$ imagine/build an exterior universe $V[G]$ (whose properties are bound to $\mathbb{P}$, the forcing relation $\Vdash_{\mathbb{P}}$, and $V$.)

From this perspective the original question can be recast as: How does one prove certain statements are preserved by a particular $\mathbb{P}$? The answer: it depends, but in general there are two methods which appear most. From here we will consider a couple examples.

**Consider the case of $AC$**: Now, $AC$ is equivalent to the statement $ \forall X\ \exists \alpha \in ON\,\ f\subset X \times \alpha\ (\ f$ is injective$)$ and we want to show that this statement is preserved between forcing extensions.

**First Method:** (Brute force combinatorics) (paraphrasing what is found in Kunens' text)

We must show $\forall p\in \mathbb{P}\ \exists q \le p\ [q \Vdash \forall \dot{X}\ \exists\check{\alpha}\in ON,\ \dot{f} \subset \dot{X} \times \check{\alpha}\ (\dot{f} $ is an injection with domain $\dot{X})]$. Using the standard facts about the forcing relation $\Vdash$, this becomes: for every $\dot{X} \in V^{\mathbb{P}}$ and $p \in \mathbb{P}$, there exists some $q \le p$, $\alpha \in ON$, and $\dot{f}\in V^{\mathbb{P}}$ such that $q \Vdash [\ \dot{f} $ is an injection with domain $\dot{X}\ ].$

In order to show this assertion, we are going to explicitly build the name $\dot{f}$, and produce the condition $q$ which forces "$\dot{f} $ *is an injection with domain* $\dot{X}$" only using statements $ZF+AC$ can prove.

To this end, note that since we are assuming $AC$, we may assume $\dot{X} = \bigcup_{\gamma\in\mu}\{ \langle \sigma_\gamma, q \rangle: q \in A_\gamma \}$, where each $\sigma_\gamma$ is a $\mathbb{P}$-name, $\mu$ is some cardinal, and $A_\gamma$ is an anti-chain such that $\forall q \le p$: we have $q \Vdash [\sigma_\gamma \in \dot{X}]$, if and only if $\{ s \in A_\gamma: s \not\perp q \}$ is maximal below $q$.

Let $\dot{g}= \bigcup_{\alpha\in\mu} \{ \langle \rho_\alpha(q), q \rangle: q \in A_\alpha \}$ where $\rho_\alpha(q) = \{\langle \sigma_\alpha, q \rangle, \langle \{\langle \sigma_\alpha, q \rangle, \langle \check{\alpha}, q \rangle, \}, q \rangle \}$. Then, taking $q=p$, $\alpha = \mu$ and $\dot{f} = \dot{g}$ establishes the result.

**Second Method:** (Sage like and Model Theoretic)

The statement "$f$ *is an injection*" is $\Delta_0$. Moreover, if $f$ is an injection, then $ZF(?C)$ proves that the same holds for all of its subsets. Noting that for every $\dot{X} \in V^{\mathbb{P}}$ there exists some $Y \in V$ such that $ 1 \Vdash \exists \dot{g} \subset \dot{X} \times \check{Y}\ (\dot{g} $ is an injection $)$ (namely $Y = \{\langle \sigma, q \rangle^\check{\ }: \langle \sigma, q \rangle \in \dot{X} \}$) and by $AC$ there exists some injection $f : Y \to \mu$ (some cardinal $\mu$), the result follows.

**Conclusion for $AC$:** The reason $AC$ is preserved when passing to a generic extension is essentially because it is equivalent to a "well-positioned" statement. Where "well-positioned" in this case means: it asserts the existence of an object with a $\Delta_0$ property and this $\Delta_0$ property is maintained by each of its subsets, all of which can be turned into a witness for some particular instance of $AC$. In this way, the truth or falsity of $AC$ in the extension does not depend on $\mathbb{P}$ and depends only on whether or not it held in $V$.

To contrast this with other weak forms of $AC$: consider $DC_\omega$. In order for $DC_\omega$ to be preserved we must avoid something. In particular we must avoid adding some $R \subset X \times X$ which is an entire binary relation and witnesses the failure of $DC_\omega$ (i.e. there is no $\{x_n:n\in\omega\}$ so that $\langle x_n, x_{n+1} \rangle \in R$) and we have no reason to expect this is the case without knowing something about $\mathbb{P}$.