TITLE: Finiteness properties of mapping class groups
QUESTION [5 upvotes]: Question: Is it known if the mapping class groups (of surfaces of finite type) are similar to Gromov-hyperbolic groups in the following senses: 
1) Does every finite generating set give us a finite presentation?
2) Are there finitely many cone types with respect to any (some) finite presentation?

Definition: A group is called hyperbolic if its Cayley graph is Gromov-hyperbolic, i.e., triangles are $\delta$-thin for some positive $\delta$.
Definition: Let $G$ be a group with a finite generating set $S$ and let $g \in G$. The cone type of $g$ w.r.t. $S$ is the following set:
$ \mathcal{C}(g) = \{ h \in G | \hspace{2mm} d(e,gh) = d(e,g)+d(e,h)\}$
,where $d(.,.)$ shows the distance in the Cayley graph w.r.t. to the generating  set $S$.

REPLY [2 votes]: It looks like question 1) and 2) has been answered, but in case you're still wondering generally "in what ways are mapping class groups similar to Gromov-hyperbolic groups?", you may be interested in reading about hierarchically hyperbolic spaces, introduced by Jason Behrstock, Mark F. Hagen, Alessandro Sisto in this paper:
Hierarchically hyperbolic spaces I:
curve complexes for cubical groups.
For a less technical overview you can also check out Sisto's blog post.
(I heard about everything here from Jacob Russel's talk at GSCAGT.)