To address your special case: threadable cardinals are not necessarily 1-reflecting. In fact, the assertion, "$\kappa$ is $\alpha$-threadable for every $\alpha$ such that $\alpha^+ < \kappa$" does not imply that $\kappa$ is 1-reflecting. To see this, suppose that $\kappa$ is a weakly compact cardinal whose weak compactness is preserved by $\kappa$-directed closed forcing (this is overkill, for the sake of concision). Now let $\mathbb{S}$ be the standard forcing notion to add a non-reflecting stationary subset to $\kappa$ (or to $S^\kappa_\omega$, or to your favorite stationary subset of $\kappa$) by initial segments. In $V^{\mathbb{S}}$, let $\mathbb{T}$ be the forcing that shoots a club in $\kappa$ disjoint from this generically added stationary set. The point is that, for all $0 < \beta < \kappa$, the two-step iteration
$\mathbb{S} * \dot{\mathbb{T}}^\beta$ (where the second iterand is a full-support product) has a dense $\kappa$-directed closed subset. Now move to $V^{\mathbb{S}}$. In this model, there is a non-reflecting stationary subset of $\kappa$, since we have just explicitly introduced one with $\mathbb{S}$. However, if $\alpha < \kappa$, then $\square(\kappa, \alpha)$ must fail. This is similar to arguments in our paper that you cited, but is basically because forcing with $\mathbb{T}$ would have to add a thread to any $\square(\kappa, \alpha)$-sequence, but such a thread cannot be added by a forcing $\mathbb{Q}$ such that $\mathbb{Q}^{\alpha^+}$ is $\kappa$-distributive. $\kappa$ is inaccessible in this model; similar arguments will work at successors of either singular or regular cardinals.
The situation becomes more interesting if you increase the threadability assumption to the failure of $\square(\kappa, < \kappa)$, which is actually equivalent to the tree property holding at $\kappa$. If $\kappa$ is inaccessible, then this is equivalent to $\kappa$ being weakly compact, in which case $\kappa$ is $\alpha$-reflecting for all $\alpha < \kappa$. If $\kappa$ is a double successor cardinal, i.e., $\kappa = \lambda^{++}$, then it is shown in Cummings-Friedman-Magidor-Rinot-Sinapova (2016, preprint) that the tree property can consistently hold at $\kappa$ while there is a non-reflecting stationary subset of $S^\kappa_{<\lambda^+}$. Perhaps somewhat surprisingly, it actually turns out to be rather difficult to simultaneously obtain reflection and the tree property at successors of small singular cardinals. This was first done in Fontanella-Magidor(2017), for $\kappa = \aleph_{\omega^2 + 1}$. The question remains open, and seemingly quite difficult, whether the tree property and stationary reflection can simultaneously hold at $\aleph_{\omega + 1}$.
I think this covers the main points regarding implications from threadability to reflection. Implications from full reflection of the form $\mathrm{Refl}(\alpha, \kappa)$ to threadability are covered, I think pretty exhaustively, in the paper with Hayut that you cited. There are some interesting (to me, at least) open questions regarding implications from $\mathrm{Refl}(\alpha, S)$ to threadability, where $S$ is some specific stationary subset of $\kappa$. For example, it is open whether $\mathrm{Refl}(\omega, S^{\omega_2}_{\omega})$ implies the failure of $\square(\omega_2, \omega).$ In a related vein, there is a recent preprint by Fuchs in which he investigates the effect of diagonal stationary reflection hypotheses on threadability.
