If you are willing to expand your final question slightly to include general graphs of groups then the answer to Question 2 is certainly “no”.
In their famous 2008 GAFA paper Special cube complexes, Haglund and Wise give a version of the Rips construction where the central group is the fundamental group of a thin VH complex. That is, for any finitely presented group $Q$, they construct a short exact sequence
$1\to K\to \Gamma\to Q\to 1$
where $K$ is finitely generated and $\Gamma$ is the fundamental group of a thin VH complex. I won't recall the definition of a thin VH complex, but will just note that any such group is the fundamental group of a graph of free groups with malnormal edge groups. If $Q$ is infinite and not free then, by a theorem of Bieri, $K$ is not finitely presented. In particular, $\Gamma$ is certainly not locally quasiconvex.
I'm sure one can also cook up an example which is an actual amalgam. For instance, if $b_1^{(2)}(G)=0$ then $G$ virtually fibres by Kielak's theorem, and hence is not locally quasiconvex. In fact, Wise has conjectured that local quasiconvexity fails whenever the Euler characteristic of $G$ is positive.
Not much is known about Question 1. However, one can give a conjectural answer in the framework of my paper 'Rational curvature invariants for 2-complexes', which is designed to address exactly these kinds of questions.
For a suitable graph of graphs $X$ representing $G$, the Howson property should be at least partially controlled by the invariant $\rho_+(X)$ that I define in the paper. For instance, Conjecture 12.9 says that if $\rho_+(X)<0$ then $G$ should be locally quasiconvex, and in particular have the Howson property.
It is equally reasonable to conjecture that, if $\rho_+(X)>0$, then $G$ should not have the Howson property. However, the case of $\rho_+(X)=0$ will be less definitive, since $\mathbb{Z}^2$ has the Howson property but $F\times\mathbb{Z}$ doesn't.
I won't recall the definition of $\rho_+(X)$ here, since it's a little complicated, but note that it is a rational number that can be determined by an explicit linear programming problem. My guess is that, in the case of graphs of graphs, the definition of $\rho_+(X)$ can be simplified. See §11.5 of the paper for some further discussion of $\rho_+$ for graphs of free groups. (And feel free to email me if you would like more details on any of this.)
Anther reference is Wise's survey article 'An Invitation to Coherent Groups', which includes some discussion of the Howson property.