Under what conditions is broadcast possible? (Cryptographically, and in the model of Maurer 2006)
This is a somewhat standard method for generalizing results based on threshold adversaries. Let $n$ be the number of parties and let $\mathcal{C} \subseteq 2^{[n]}$ denote the family of subsets that the adversary can corrupt. For example, in the case of honest majority, $\mathcal{C} = \{ C \subseteq [n] : |C| &lt; n/2 \}$. 


$\mathcal{C}$ has the Q2 property if no two corruptible sets $A, B \in \mathcal{C}$ satisfy $A \cup B = [n]$. In Maurer's notation, $[n] \not\in \mathcal{C} \sqcup \mathcal{C}$.
$\mathcal{C}$ has the Q3 property if no three corruptible sets $A, B,C \in \mathcal{C}$ satisfy $A \cup B \cup C = [n]$. In Maurer's notation, $[n] \not\in \mathcal{C} \sqcup \mathcal{C} \sqcup \mathcal{C}$.


Generally speaking, classical results that require honest majority will generalize to any Q2 corruption pattern. Results that require honest two-thirds majority generalize to Q3.

I think the first appearance of this generalization is in Complete Characterization of Adversaries
Tolerable in Secure Multi-Party Computation by Hirt &amp; Maurer. It's generalized even farther in MPC vs. SFE: Perfect Security in a Unified Corruption Model by 
Beerliova-Trubiniova et al. They consider an adversary that can corrupt any $S \in \mathcal{P}$ passively, or corrupt any $S \in \mathcal{A}$ actively, or corrupt any $S \in \mathcal{F}$ in a fail-stop way. These results are harder to state in a stackexchange answer.

Regarding the question about broadcast, the actual tight bound depends very heavily on the model (poly-time vs information-theoretic, [im]perfect security, PKI setup). Also agreement, consensus, and broadcast are all slightly different. But in general, information-theoretic broadcast with no setup requires honest 2/3rds majority (Q3 corruptions).