Edit Jul 25: These results may be strengthenable by using theorem 7.8 of chapter V of Admissible Sets and Structures instead of lemma 1, I may edit this post in the future with any resulting improvements.
$\newcommand{\vrecv}[1]{\vert #1\textrm{-rec.}\vert}\newcommand{\vrev}[1]{\vert #1\textrm{-r.e.}\vert}\newcommand{\vpfv}[1]{\vert #1\textrm{-pf}\Sigma_1\vert}$Let $\vrecv\alpha$ be short for [1]'s $\vert\alpha\textrm{-recursive}\vert$, i.e. the sup of the order types of $\alpha$-recursive relations (i.e. $\Delta_1$-on-$L_\alpha$ with parameters.) $\vrev\alpha$ is as in [1], the same as $\vert\alpha\textrm{-recursive}\vert$ but for $\alpha$-r.e. relations (i.e. $\Sigma_1$-on-$L_\alpha$ with parameters.) Let $\vpfv\alpha$ be the same but for parameter-free-$\Sigma_1$ definable relations. Call $M$ $\Sigma_1$-pointwise definable if every member of $M$ is definable in $M$ by a $\Sigma_1$ formula (vis. [2]).
(A remark, $\delta_{\alpha+1}$ is not defined as $\vpfv{\delta_\alpha}$, $\delta_\alpha$'s definition differs from Aguilera's $\eta_\alpha$ in this respect. But they are equivalent, a candidate for $\delta_{\alpha+1}$ is $\vpfv{\delta_\alpha}$ by letting $\beta=\delta_\alpha$, and it is the least candidate since any $\delta_\alpha<\gamma<\vpfv{\delta_\alpha}$ won't be equal to $\vpfv\beta$ for a $\beta<\alpha$, as $\vpfv\beta$ would be $\delta_\alpha$.)
Barwise's Admissible Sets and Structures, theorem V.7.8: Let $\alpha$ be admissible and $\gamma<\alpha$ be any ordinal, and $\beta<\alpha$ be least such that $L_\beta\prec_{\Sigma_1}L_\alpha$ and $\beta>\gamma$. $L_\beta$ is the set of $x\in L_\alpha$ that are definable in $L_\alpha$ by a $\Sigma_1$ formula with parameters that are ordinals $<\gamma$.
Remark: This seems similar to results about pointwise definability of various stable levels of $L$ in [2]. According to [5], the admissibility assumption can be replaced with $\alpha$ being a limit ordinal $>\omega$.
Corollary 1: If $\alpha>\omega$ is a limit ordinal such that there is no $\sigma<\alpha$ where $L_\sigma\prec_{\Sigma_1}L_\alpha$, then $L_\alpha$ is $\Sigma_1$-pointwise definable.
Lemma 2: Let $\alpha$ be as in corollary 1. Then $\vrecv\alpha\le\vpfv\alpha$. If additionally we have $\alpha$ admissible, then $\vrecv\alpha=\vpfv\alpha$.
Proof: Any relation $\Delta_1$-definable on $L_\alpha$ with parameters $\vec x$ is $\Sigma_1$-definable on $L_\alpha$ with parameters. It is then also $\Sigma_1$-definable without parameters by conjoining theorem V.7.8's $\Sigma_1$ definitions of $\vec x$. If $\beta<\vrecv\alpha$ is such that there is an $\alpha$-recursive w.o. $R$ of $\alpha$ of order type $\beta$, $R$ is also parameter-free-$\Sigma_1(L_\alpha)$, so we have that $\vrecv\alpha\leq\vpfv\alpha$. In the additional case that $\alpha$ is admissible, by (3) of [1, p.179] we have that $\vrecv\alpha=\vpfv\alpha$. $\square$
Say that $\alpha$ has property (2) if $\alpha$ is as in lemma 2 ($\alpha>\omega$ is a limit ordinal and there is no $\sigma<\alpha$ s.t. $L_\sigma\prec_{\Sigma_1}L_\alpha$.)
Positive answer for question #1: Any $\alpha$ where $\delta_\alpha$ is uncountable is a positive answer to question #1.
Proof: If $\delta_\alpha$ is uncountable, since $\delta_{\alpha+1}\geq\delta_\alpha\cdot 2$, there are uncountably many ordinals between $\delta_\alpha$ and $\delta_{\alpha+1}$. However there are only countably many $\delta_\alpha<\beta<\delta_{\alpha+1}$ which can be the order type of a parameter-free $\Sigma_1(L_{\delta_\alpha})$ w.o., since there are countably many $\Sigma_1$ formulae. $\square$
The following helps bound the second part of question #1, asking about the least positive answer $\delta_\alpha$.
Partial result for second part of question #1: For any $\alpha$ such that $\delta_\alpha$ has property (2), there is no ordinal $\delta_\alpha<\beta<\delta_{\alpha+1}$ as in question #1.
Proof: Assume towards a contradiction that there are positive answers $\alpha,\beta$ from question #1 such that $\delta_\alpha$ has property (2). Then $L_{\delta_\alpha}$ is $\Sigma_1$-pointwise definable. As $\vpfv{\delta_\alpha}=\delta_{\alpha+1}>\beta$, take some parameter-free-$\Sigma_1(L_{\delta_\alpha})$ well-ordering $R$ of $\delta_\alpha$ that has order type $>\beta$. Take the $b\in L_{\delta_\alpha}$ that is the point at rank $\beta$ in $R$, use $\Sigma_1$-pointwise-definability of $L_{\delta_\alpha}$ to obtain a $\Sigma_1$ definition of $b$ in $L_{\delta_\alpha}$, and define a new well-ordering that is $R$ cut at $b$. This ordering is parameter-free-$\Sigma_1(L_{\delta_\alpha})$-definable and has order type $\beta$, contradiction. $\square$
Corollary: No $\delta_\alpha$ below the least p.f.e.c. which does not have property (2) can be a positive answer to question #1.
Partial result for question #2: If $\delta_\alpha$ is good, and $L_{\delta_\alpha}$ is $\Sigma_1$-pointwise-definable, then $\vpfv{\delta_\alpha}=\delta_\alpha^+$, i.e. $\delta_{\alpha+1}=\delta_\alpha^+$. Additionally for any $\delta_\alpha<\beta<\delta_\alpha^+$ there is a parameter-free-$\Sigma_1(L_{\delta_\alpha})$ well-ordering of $\delta_\alpha$ of order type $\beta$.
Proof: Let $\delta_\alpha<\beta<\delta_\alpha^+$ be arbitrary. Let $R$ be a $\delta_\alpha$-recursive well-ordering of $\delta_\alpha$ of order type $\beta$, one exists by badness of $\delta_\alpha$. Since $L_{\delta_\alpha}$ is $\Sigma_1$-pointwise-definable, conjoin the definitions of any necessary parameters to obtain a parameter-free-$\Sigma_1(L_{\delta_\alpha})$ definition of $R$. This constructs the desired parameter-free-$\Sigma_1(L_{\delta_\alpha})$ well-ordering. As $\delta_\alpha<\beta<\delta_\alpha^+$ was arbitrary, this shows $\vpfv{\delta_\alpha}=\alpha^+$. $\square$
Remark for #2: All $\xi$ less than $\sigma_1^1$ (least $\Sigma_1^1$-reflecting ordinal) are locally countable and good (see [1]), and there are club-in-$\sigma_1^1$-many such $L_\xi$ that are $\Sigma_1$-pointwise-definable, so this gives many $\delta_{\alpha+1}<\sigma_1^1$ that are admissible. The case for limits of admissibles is that $\delta_\alpha=\textrm{sup}\{\delta_\beta\mid\beta<\alpha\}$, since $\delta_\alpha$ may not be less than any $\delta_\beta$ with $\beta<\alpha$, and limits of p.f.e.c. are p.f.e.c.
For this approach to provide answers for questions #3 and #4, each of the two conditions of (2) must be weakened. For #3 the admissibility assumption in lemma 2 would have to be weakened, as for any positive answer $\gamma$ for $\alpha>1$, $\gamma<\vpfv\gamma<\gamma^+$. For #4 the "no $\sigma<\alpha$ s.t. $L_\sigma\prec_{\Sigma_1}L_\alpha$" assumption would have to be weakened, so that work can be done with parameter-free-$\Sigma_1(\delta_{\alpha+1})$ relations while $L_{\delta_\alpha}\prec_{\Sigma_1}L_{\delta_{\alpha+1}}$.
One thing relevant to question #4:
Lemma 3: If $\alpha$ is the least ordinal that is $\Sigma_1^1$-reflecting on $\Sigma_1^1$-reflecting ordinals, $\alpha=\delta_\alpha$.
Proof: Since $\alpha$ is a limit ordinal, $\alpha=\textrm{sup}\{\delta_\xi\mid\xi<\alpha\}$. (cf. remark for #2.) If $\beta<\alpha$ is bad, then $\vrecv\beta<\beta^+<\alpha$ is a p.f.e.c. So it is sufficient to show that $\alpha$ is the $\alpha$th bad ordinal, or since all ordinals $<\alpha$ are locally countable, that $\alpha$ is the $\alpha$th $\Sigma_1^1$-reflecting ordinal. $\alpha$ is a limit of $\Sigma_1^1$-reflecting ordinals, so for any $\Sigma_1^1$-reflecting $\xi<\alpha$, $\xi+\omega+1<\alpha$, ensuring finite sequences from $L_\xi$ are in $L_\alpha$.
"$\gamma$ is $\Sigma_1^1$-reflecting" iff $\forall\vec x\in L_\gamma\forall(\ulcorner\phi\urcorner\in\Sigma_1)(L_{\gamma+1}\vDash\phi(\vec x)\rightarrow\exists\delta<\gamma(L_{\delta+1}\vDash\phi(\vec x)))$, which by work in [3] can be seen to be $\Sigma_1$ on $L_\alpha$. The set $X=\{\gamma<\alpha\mid\gamma\textrm{ is }\Sigma_1^1\textrm{-reflecting}\}$ is $\Sigma_1(L_\alpha)$, so its enumerating function is $\Sigma_1(L_\alpha)$. $\alpha$ is $\Pi_2$-reflecting and therefore admissible [4], and no $\Sigma_1(L_\alpha)$ function can map an ordinal $<\alpha$ cofinally into $\alpha$, so $\textrm{dom}(\textrm{enumerating function of }X)=\alpha$. $\square$
[1]: R. Gostanian, "The Next Admissible Ordinal", Annals of Mathematical Logic vol. 17, 1979
[2]: W. Marek, "Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts", Fundamenta Mathematicae vol. 82, 1974
[3]: A. Lévy, A hierarchy of formulas in set theory, Memoirs of the American Mathematical Society iss. 57, 1965
[4]: W. Richter, P. Aczel, "Inductive Definitions and Reflecting Properties of Admissible Ordinals", in Studies in Logic and the Foundations of Mathematics vol. 79, 1974
[5]: M. Möllerfeld, M. Rathjen, "A note on the $\Sigma_1$ spectrum of a theory", Archive for Mathematical Logic vol. 41, 2002
