$\newcommand\ZFC{\text{ZFC}}
\newcommand\HOD{\text{HOD}}
\newcommand\P{\mathbb{P}}
\newcommand\Q{\mathbb{Q}}$
I have found a positive answer to question 1.

**Theorem.** Every model of $\ZFC$ has a forcing extension
satisfying $V\neq\HOD$, in which every $\Sigma_2$-definable set
has an ordinal-definable element.

Proof. Start in $V$ as a ground model. Without loss, by forcing if
necessary, we may assume that $V$ satisfies $V=\HOD$, so that
there is a definable well-ordering of the universe. Enumerate the
$\Sigma_2$ formulas $\varphi_0,\varphi_1,\ldots$. Note that
we may refer to $\Sigma_2$-truth since there is a universal truth
predicate for truth of bounded complexity (so there will be no
issues with Tarski's theorem on the non-definability of truth). We
define a forcing iteration $\P$ of length $\omega$. At the first
stage, we consider the formula $\varphi_0$. Let $\theta_0$ be any
fixed cardinal, and ask: is there a $\leq\theta_0$-closed forcing
extension $V[g_0]$ in which $\varphi_0$ defines a nonempty set
$A_0$? If so, we perform such a forcing (choose the least poset
forcing this), and let $\lambda_0$ be the smallest $\beth$-fixed
point above the size of that forcing and $\theta_0$ so that also
$\varphi_0$ is witnessed in $V_{\lambda_0}^{V[g_0]}$. Next,
perform additional $\leq\lambda_0$-closed forcing over $V[g_0]$ to
an extension $V[g_0][h_0]$, where $h_0$ forces to code one of the elements of
$A_0$ into the GCH pattern above $\lambda_0$. This preserves the
definition of $A_0$ by $\varphi_0$, while ensuring that $A_0$ has
an ordinal definable element. Now, let $\theta_1$ be well above this coding, and continue.

At stage $n$, we have the partial extension
$V^{(n)}=V[g_0][h_0]\cdots[g_{n-1}][h_{n-1}]$, which performed
forcing below $\theta_n$. We ask whether we can perform
$\leq\theta_n$-forcing so that $\varphi_n$ defines a nonempty set
$A_n$ in the extension. If so, we do that forcing, let $\lambda_n$
be large enough to witness the $\Sigma_2$ property for
$\varphi_n$, and then perform GCH coding above that so as to make
an element of $A_n$ ordinal-definable, and let $\theta_{n+1}$
larger than all that. (Otherwise, we ignore $\varphi_n$ and let
$\theta_{n+1}=\theta_n$.)

Consider the corresponding extension $V[G]$, where $G\subset\P$ is
$V$-generic, and we use full support in $\P$. Finally, we force to
add a Cohen subset $H\subset\delta$, where $\delta$ is a regular
cardinal above $\sup_n\theta_n$, since this will force $V\neq\HOD$ in $V[G][H]$. The desired model is $V[G][H]$.
Because we used full support, it follows that the tail forcing in
$\P$ after stage $n$ is $\leq\theta_n$-closed, as is the forcing
to add $H$, and so preserves sets of rank below $\lambda_n$. Thus,
if $\varphi_n$ defines a nonempty set in $V[G][H]$, then it would
have done so in some sufficiently closed extension of the model at
stage $n$, and we would have forced to add an ordinal-definable
element to that set, afterwards preserving the definition of that
set and of the ordinal-definable element.

Thus, $V[G][H]$ is a model of $V\neq\HOD$, as the forcing to add
$H$ is weakly homogeneous, such that every $\Sigma_2$-definable
nonempty set has an ordinal-definable element. QED

Meanwhile, Emil's idea in the comments seems to lead to a negative answer
to question 2. Perhaps he will post a fuller account of it (or I can later if he doesn't care to).