Let me preface this by stating the obvious in that there is no injection from a given uncountable cardinal to any countable ordinal, thus any system of describing "analogous properties" of large countable ordinals relative to an uncountable set of ordinals will be flawed in infinitely many ways. The connection between large cardinal axioms and large unrecursive countable ordinal axioms is based on intuition, not that there is anything wrong with that. The bottom line is, "recursively greatly Mahlo" ordinals can all be stable, or nonprojectible or $\Pi_3$-reflecting if you define them as such, but they absolutely don't have to be.
I suggest reading "Generalizing the Mahlo hierarchy, with applications to the Mitchell models" by Baldwin, where he defines a very structured hierarchy of Mahloness in analogy to the Mitchel rank for measurability of cardinals. I myself have not read more than a few pages but we only need to use the outline from the paper, so the descriptions I will give here are very oversimplified and do not constitute definitions, so keep that in mind. $m(\kappa)$ is a rank of Mahloness that essentially tells the that a filter on the stationary subsets of $\kappa$, if applied less than $m(\kappa)$ times to an equivalence class on $\kappa$
based on stationarity, will result in a set that is neither 0 nor the set of all nonstationary subsets of $\kappa$. This ranking does not provide much new insight for the structure of Mahloness as $m(\kappa)\leq\kappa^+$ is always the case by definition and for every greatly Mahlo $\kappa$ we already have $m(\kappa)=\kappa^+$ and vice versa, thus this notation is useless for trying to understand cardinals that go beyond greatly Mahlos in stationary properties that are themselves not on the level of the hierarchy of weakly and supercompacts. Further down the paper in section 3, functions $\bar m(\kappa), m^*(\kappa)$ are defined with more complex definitions that provide further structure to the different types of Mahlo cardinals and the different levels of iterated stationarity. According to Baldwin, for these to work as desired, we must assume a particular type of failure of GCH in $2^\kappa=2^{\kappa^+}$ at least for greatly Mahlo $\kappa$, that is, $|\mathcal P(\kappa)|$, which is the size of the domain and codomain of $m(), \bar m(), m^*()$, must be greater or equal to a limit of regular cardinals above $\kappa$. There is no analogy of this that can be translated down to the constructible universe of countable sets $L_{\omega_1}$, which is where we want the least "recursively greatly Mahlo" to fall.
Let $M[\beta](0)$ be the first recursively $\beta$-Mahlo ordinal and let $M[\beta](\lambda+\alpha)$ be the $\alpha$th recursively $\beta$-Mahlo ordinal above $M[\beta](\lambda)$ whenever $\alpha < M[\beta+1](0)$ and let $\beta$-Mahlo ordinals be exactly those ordinals $\kappa$ such that $L_\kappa\models$KP+"$\forall\gamma<\beta:$ the $\gamma$-Mahlo ordinals form a stationary subset of the ordinals".
We could define a collapse of $M[\beta+1](\alpha)$ inside $M[\beta](\text{___})$ to form an evaluation for the hierarchy of $\beta$-Mahlos, limits of such, inaccessible limits of such, hyper-inaccessible limits of such, hyper-hyper-inaccessible limits of such and so on, and keep going from there. If we use, say, the least $\Pi_3$-reflecting ordinal $K$ to collapse inside of $M[\text{___}](0)$ such that $M[K+\gamma](\beta)$ is the $(1+\beta)$th ordinal $\alpha$ that is $\alpha$-Mahlo and for all $\delta_0\in\gamma\land\delta_1\in\alpha:$ the ordinals of the form $M[K+\delta_0](\delta_1)$ form a stationary subset of $\alpha$ with respect to clubs definable in $L_{\omega_1}$. We can have an ordinal representation system $M[K+K\alpha+\beta](\gamma)$ being the $(1+\gamma)$th $\beta$-hyper$^\alpha$-Mahlo ordinal and so on. Then finally, we can define the recursively greatly Mahlo ordinals as exactly those that are of the form $M[\omega^{\rm CK}_{K+1}](\alpha)$ for some $\alpha$ as $\omega^{\rm CK}_{K+1}$ is admissible and above $K$ and thus not reacheable from below by some recursive well-ordering on $K+1$ and thus the least recursively greatly Mahlo will be greater than the $M$-collapse of any ordinal in $[K,K^+)$ or even simply in $K^+$.
If $\alpha^+$ is the least regular above $\alpha$, that means that there is no surjection from $\alpha$ to $\alpha^+$. Analogously, if $\alpha^+$ is the least admissible above $\alpha$ then that means there is no surjection from $\alpha$ to $\alpha^+$ that consitutes a recursive or hyperarithmetical reordering of $\alpha$. This is not a definition but it suffices for making the analogy.
I believe this way of defining the recursively greatly Mahlos fits the bill for placing them in the hierarchy of recursively Mahlos in such a way that they always have the recursively $\beta$-hyper$^\alpha$-Mahlo ordinals and all kinds of limits of such be unbounded in them, yet without the greatly Mahlos necessarily having $\Pi_3$-reflecting or stronger properties.
The first chromatic cardinal is the first Mahlo cardinal. (Per the connection with the reflection in the question, I assume that in the definition, $α$ and the first argument of $c_i$ need not be inaccessible.) If we allowed $c_i$ for $i≤δ$, then the first chromatic cardinal would be the first Mahlo above $δ$.
If $κ$ is Mahlo, then it is chromatic. Define club $C⊆κ$ with $λ∈C ⇔ ∀α<λ \, ∀i \, \min(\{β>α:c_i(α,β)\})<λ$ where $\min(S)$ is modified to return 0 if $S$ is empty. Then any inaccessible $λ∈C$ satisfies the desired property.
In the other direction, if $κ$ is below the first Mahlo, then there is a one-to-one function that for every inaccessible below $κ$ returns a smaller ordinal. For example, pick a club $D$ that excludes inaccessibles. If $β$ is the least inaccessible above some element of $D$, then set $f(β)=\max(β∩D)$. Let $λ=\min(D\setminus β)$, and if there are inaccessibles in $(β,λ)$, let $λ'$ be largest inaccessible or limit of inaccessibles $<λ$. Set (for example) $f(λ')=β$, and analogously to $D$ pick a club $E⊂λ'$ above $β$ that excludes inaccessibles. Analogously proceed by recursion until all inaccessibles are taken care of.
For your second question, despite the power of second order logic, the cardinals in question are precisely inaccessible $α$ with $V_α ≺_{Σ_{1,V}} L_1(V_{α+1})$, where $Δ_{0,V}$ formulas allow querying about $x$ whether $∃α \, x=V_α$ (for the true $V$), and $L$ is the constructible hierarchy. $Σ_{1,V}$ formulas are upwards absolute, and $L_1$ corresponds to definability, so the above reduces to the desired reflection relation. If there is an inaccessible $λ$ with $V_λ ≺_{Σ_2} V$, then $α$ for the question exists for every parameter-free definable $Δ^V_2$ logic (even if there are no Mahlos in $V$; also, we can allow parameters in $V_λ$).
Best Answer
The paper "Small Definably-large Cardinals" by Roger Bosch proves that an inaccessible cardinal is $\Sigma_{n+1}$-Mahlo if and only if it is $\Pi_n$-Mahlo except for $n=1$ (I'm referring to the boldface hierarchy, not the lightface hierarchy which is less relevant to your question) and that an inaccessible cardinal is $\Sigma_2$-Mahlo if and only if it is $\Delta_2$-Mahlo. Whether $\Pi_1$-Mahlo cardinals are always $\Delta_2$-Mahlo (and thus $\Sigma_2$-Mahlo) an open problem as far as I know.