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. 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 $\beta < \kappa$, 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)][1] that the tree property can consistently holds 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)][2], 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][3] in which he investigates the effect of diagonal stationary reflection hypotheses on threadability.


  [1]: http://papers.assafrinot.com/paper30.pdf
  [2]: https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/div-classtitlereflection-of-stationary-sets-and-the-tree-property-at-the-successor-of-a-singular-cardinaldiv/1D1D6FF82A157E1373A14C1E4CC94233
  [3]: http://www.math.csi.cuny.edu/~fuchs/DiagonalReflectionsOnSquares.pdf